Statistics.
Cyntha Struthers
Section 001
Email: castruth@gmail.com
Mondays/Wednesdays/Fridays 1:30pm2:20pm in STP 105
This is the second time I'm taking this course. The first time around, I received an injury about a month in that made going to this class no longer feasible. This set of notes therefore contains the Spring 2016 version of the course, combined with the Winter 2017 version of the course. Over time, these will be merged together.
Course notes are available on LEARN  don't buy the paper copy unless you really want to.
There are iclicker questions, worth 5% overall. Lowest 25% of clicker answers are not included in grade calculation. There are also 5 assignments, each worth 1%, that teach the use of R. Some tutorials include tests, and there are two midterms (each worth 15%). Final exam worth 50%.
In STAT230, we were given information about the population, and wanted to figure things out about individual samples. In STAT231, we will be learning about the opposite  given information about a sample of the population, what can we figure out about the population?
An empirical study is a study in which we learn by observation or experimentation. Empirical studies involve uncertainty by nature  if we run the same experiment multiple times, we are very likely to get results that aren't totally identical. Using the probability tools we learned about in STAT230, we can model this uncertainty and use it to learn more from the study results.
A population is a collection of units, like all students taking STAT231, or all the cars on a particular road. A process is a system by which units are produced, like a car insurance claim system (it produces car insurance claims as units).
Both of these are collections of units, but processes generally emit units over time, while populations are generally all already there at a certain point in time. For example, a population might be "current STAT231 students", while a process might be "past, present, and future STAT231 students".
Read chapter 1 and do the end of chapter problems to review STAT230. Assignment 1 due next Monday, concerns the setup and use of R.
A variate is a characteristic of units in a population/process, usually represented as \(x\), \(y\), or \(z\). They fall under a couple types (the type of the variate determines what kind of operations we can perform on them):
Knowing the type of a variate helps us choose a good probability model for the data.
An attribute of a population/process is a function of a variate that is defined for every unit in the population (not every unit in a study or survey). For example, if we have all students in a class (the population), and each individual student (the unit in the population) has a grade (variate on every unit), then an attribute might be the class average or proportion of students who passed the class (a function of the grade variate). Common attributes are mean/median/mode/variance/proportion.
Empirical studies are either observational or experimental. Observational empirical studies are those in which information is collected about the population/process without attempting to change any variates of units in the population. When we do change variates, it is an experimental empirical study. Generally, in an experimental empirical study, researchers will do something to one subset of the population, and leave the rest as a control group. Note that even in an observational empirical study, researchers can change variates outside of the context of the study  for example, researchers might ask the entire population to complete a survey, which changes variates like "how fed up is this participant with surveys?", yet this is still an observational study, since the survey is external to the study itself.
Sample surveys are observational empirical studied in which we select a representative sample of units from population and determine relevant variates for those units.
Numerical and graphical summaries are used to make sense of large datasets, where we would otherwise not be able to by looking at the numbers directly.
Some numerical summaries include measures of location (sample mean, median, mode), variability/dispersion (sample variance, sample standard deviation, range, interquartile range), and shape (sample skewness, sample kurtosis).
A dataset is a set \(y = \set{y_1, \ldots, y_n}\). If \(y_i \in \mb{R}\), sample mean is \(\overline y = \frac 1 n \sum_{y_i \in y} y_i\), sample median is \(\hat{y} = \frac{y_{\left(\floor{\frac n 2}\right)}}{y_{\left(\ceil{\frac n 2}\right)}}\), and sample mode doesn't usually have a widely accepted notation.
The median of a dataset is not unique  a dataset with an even number of elements can have two medians, though usually we just average them to get a single summary value.
The mode is sometimes considered to be multiple values as well when there are several values that are most common, though in this course we consider the mode to not exist when there's no single most common value. Also, the most commonly occurring group/class in grouped data is known as the sample modal group/class.
An order statistic is just a sorted version of a dataset. It's denoted \(\set{y_{(1)}, \ldots, y_{(n)}}\).
There's actually a shortcut to get to STP from MC  go left instead of straight and cut through St. Jerome.
Tutorial test next wednesday. One question might be, "a study was conducted in October 2016 on people recovering from heart disease at a particular clinic, where they were asked to complete a questionnaire  what is the process or population?". Any reasonable answer will work: "population of people who attended the clinic in October 2016". Also look into questions like "how does the mean/median/variance change if we do something to the dataset, like add 5 to eah sample?".
A bimodal distribution is one that has what appears to have multiple "humps" on its histogram. A unimodal distribution is one that looks like a single "hump"  a bellcurvelike shape.
Common numerical summaries we care about are:
We also often care about sample skewness  how nonsymmetric the data is. This measures how different the dataset looks when comparing the side of the mean, mirrored over to the right side. Leftskewing/negativeskewing datasets have distributions that seem to "lean to the right" (they have a long left tail), while rightskewing/positiveskewing distributions seem to "lean to the left" (they have a long right tail).
One way to estimate skewness is to compare the mean and median, measuring \(\text{mean}  \text{median}\). If the mean is less than the median, then we'd say the dataset might be left skewed (more of the weight is on the left), if they were equal, we say the dataset is probably symmetric (weight is about the same above and below the mean), and if the mean was greater than the median, then we say the dataset might be rightskewed (more of the weight is on the right side).
The real measure of skewness is \(\frac{\frac 1 n \sum \left(y_i  \overline y\right)^3}{\left(\frac 1 n \sum \left(y_i  \overline y\right)^2\right)^{\frac 3 2}}\). We won't be expected to memorize this for the course. The numerator is the the mean cubed error, and the denominator somewhat resembles the population variance. The use of cubing in the numerator preserves the sign of the error, while the denominator has the \(\frac 3 2\) exponent in order to scale the numerator to a reasonable range.
A long right tail in a dataset distribution means a positive skewness. A long left tail means a negative skewness. A symmetric distribution has a skewness close to 0. A skewness between 1 and 1 is generally considered "close to 0" for Guassian data.
For example, if a person is diagnosed with a disease that has a mean time of death about 8 months after diagnoses, and survives for several months after diagnosis, then they should hope that the distribution of deaths is more rightskewed, so that most of the probability of death is already behind them.
Skewness is defined as \(\frac{\mu_3}{\sigma^3}\), where \(\mu_3\) is the third central moment, \(E[(Y  \mu)^3]\), and \(\sigma\) is the standard deviation.
Sample kurtosis is the measure of how frequent extreme observations are with respect to the normal distribution  how "s". Basically, it checks if the tails of the dataset's distribution are fatter (more extreme observations) or thinner (less extreme observations) than a normal distribution. It can also be said to measure how peaky/pointy the distribution's curve is  too much stuff in the middle means big values greater than 3, too little stuff in the middle means small values less than 3.
Kurtosis is defined as \(K = \frac{\frac 1 n \sum \left(y_i  \overline y\right)^4}{\left(\frac 1 n \sum \left(y_i  \overline y\right)^2 \right)^2}\). Note that \(K\) must be nonnegative. A normal distribution has \(K = 3\). If \(K > 3\), then the tails of the dataset's distribution are fatter than a normal distribution (there are more extreme observations), and if \(K < 3\), then the tails of the dataset's distribution are thinner than a normal distribution (there are fewer extreme observations).
Kurtosis of 0 means the data is all the same  there's no error. The uniform distribution has a kurtosis close to 1.8, because it has no outliers  all of the values are within a couple standard deviations. Values between 2 and 4 are generally considered "close to 3" for Guassian data.
Kurtosis is especially useful in finance, where accurate assessment of risks and expected returns are important. It basically tells us how normallike our data is  if the kurtosis is very close to 3, we can often assume that the dataset is normally distributed, which is extremely useful for doing statistics with.
Kurtosis is defined as \(\frac{\mu_4}{\sigma^4}\), where \(\mu_4\) is the fourth central moment, \(E[(Y  \mu)^4]\), and \(\sigma\) is the standard deviation.
So, to tell if a Guassian model is suitable for a given dataset, we can check if the mean/median are roughly the same, the skewness is close to 0, the kurtosis is close to 3, and about 95% of samples are within 2 standard deviations.
The fivenumber summary of any dataset is a tuple containing the min, first quartile, median, third quartile, and the max. They're useful for summarizing location and spread, but they can't represent things like bimodal distributions.
A frequency histogram has rectangles where the height is the number of observations in that bucket. Note that the bucket sizes might not be the same. A relative frequency histogram is a frequency histogram where the rectangle heights are scaled based on the bucket width and the total number of observations  \(\text{rectangle height} = \frac{\text{number of observations in bucket}}{\text{total number of observations} \times (\text{upper bound of bucket}  \text{lower bound of bucket})}\).
Relative frequency histograms are useful because we can superimpose a probability density function on top of them, since the Y axes match. The sum of the rectangle areas also sum up to 1 (the heights of the rectangles don't necessarily sum up to 1, however).
Potential test question: which bucket of a relative frequency histogram does the median fall into? Each percentile of a histogram lies within the first bucket whose value, when added to all previous bucket values, is greater than that percentile. For example, to find the median's bucket, we add bucket heights from left to right until we reach or exceed 0.5.
Empirical CDFs are to CDFs as relative frequency histograms are to PDFs  they're a way to let us graphically compare CDFs to the actual shape of the data. An empirical CDF is a step function that jumps up by \(\frac 1 n\) (where \(n\) is the number of observations) for each observation, where the X axis is observation values and the Y axis is the cumulative probability. Each percentile can be found in the empirical CDF by taking the X axis value correponding to that percentage of the Y axis. The CDF technically allows us to reproduce the entire dataset, given that it's drawn with enough resolution. The distribution and skewness of the data can be found by looking at the spacing between steps on the X axis  closer spacing on the left side compared to the right side implies the data tends to be positively skewed.
Empirical CDFs are often used in research papers to summarize a dataset, since most datasets are too large to include in the paper.
A boxandwhiskers plot is a useful way to show the five number summary that also highlights outliers. It's a 1D plot that looks something like this:
o < each outlier is plotted as a point, while nonoutliers are only summarized by the box and whiskers
 < top whisker represents maximum value that isn't an outlier

 < top of box represents third quartile
 
 < line in the box represents median
 
 < bottom of box represents first quartile

 < bottom whisker represents minimum value that isn't an outlier
< there might not be any outliers in the dataset
An outlier is an observation \(x_i\) such that \(x_i > Q_3 + 1.5 IQR\) or \(x_i < Q_1  1.5 IQR\), where \(Q_1, Q_3\) are the first and third quartiles, and \(IQR\) is the interquartile range \(Q_3  Q_1\). The reason we use 1.5 is because it is convention, and fits the normal distribution relatively well. In other words, an outlier is any point that doesn't fall into the range \([2.5Q_1  1.5Q_3, 2.5Q_3  1.5Q_1)]\). When we're drawing a box plot, we figure out the quartiles, figure out the ranges in which observations are outliers, and then find the max/min observations that aren't outliers.
Box plots visually show the shape of the data. A box with whiskers that are different lengths is skewed to one side, while the height of the box shows the dispersion.
Box plots tell us the approximate variance, the quartiles, the outliers, and the skewness.
Order statistics are the sorted observation values.
Consider a dataset containing mortality percentages from cancer types for female humans in Ontario, including lung cancer, stomach cancer, and other. Clearly, the population is human females in Ontario, and the dataset is categorical since the types of cancer are discrete and not ordered.
What kind of graphical summary should we use? News sites might use a pie chart, while a better option might be a bar graph (a bar graph looks a lot like a histogram, but we space out the bars to show that each bar is a different thing).
Graphs should always have appropriate axis bounds, have clearly labelled axes and titles with units, and should only be used when really appropriate.
All of the above summaries are for univariate data. However, bivariate datasets (datasets of the form \(\set{\tup{x_1, y_1}, \ldots, \tup{x_n, y_n}}\)) are also common. We often graphically summarize these datasets with a scatter plot, which is very useful for showing relationships between the two variates.
The sample correlation is a measure of association between two numerical variables. It's defined as \(r_{xy} = \frac{\sum \left(x_i  \overline x\right)\left(y_i  \overline y\right)}{\sqrt{\sum \left(x_i  \overline x\right)^2} \sqrt{\sum \left(y_i  \overline y\right)^2}}\). Basically, it's a simplified version of \(r_{xy} = \frac{S_{x, y}}{\sqrt{S_x S_y}}\), where \(S_{x_1 \ldots, x_n} = \frac 1 n \sum (x_i  \overline x)(y_i  \overline y)\) is the sample covariance and \(S_x\) is the sample variance. Note that \(1 \le r_{xy} \le 1\)  values close to 1 means that there's a linear positive correlation, and values close to 0 means that there's almost no correlation. Specifically, sample correlation measures linear association.
The sample correlation value is generally not very intuitive to guess  it's easy to find datasets that intuitively have no correlation, that have a large sample correlation. We usually wantto see a scatter plot or something to help verify our results.
An explanatory variate is a variate that's used to explain or determine the distribution in a study. A response variate is a variate that changes in response to the explanatory variate. In other words, the explanatory variate is sort of like the cause, and the response variate is sort of like the effect. For example, if we have a study that asks if clicker grades determine the final STAT231 grade, an explanatory variate might be clicker grades, while the response variate might be the final STAT231 grade. Likewise, if we have a study that asks if final grades explain clicker marks, the final grade would be the explainatory variate, and the clicker grade would be the response variate.
Start reading chapter 2. Look up information about tutorial test 1 on Odyssey and on LEARN. Seats are assigned.
Scatter plots work for numeric datasets, like discrete or continuous ones. However, we can't use them for categorical datasets, like the dataset consisting of pairs containing whether a particular student is from Canada, and which university program they're in. For these bivariate categorical datasets, we have different summaries. One of them is a table.
Another way is the relative risk. Suppose we have a dataset \(D = \set{\tup{x_1, y_1}, \ldots, \tup{x_n, y_n}}\) with two variables. Relative risk a way of summarizing bivariate categorical data  a measurement of how two variables affect each other. Basically, relative risk is \(\frac{P(A \mid B)}{P(A \mid \neg B)}\)  the probability of \(A\) occurring given that \(B\) occurrs over the probability of \(A\) occurring given that \(B\) does not occur. For independent variables, relative risk is 1. The farther the relative risk is from 1, the more strongly it implies that the variables are associated.
For example, suppose we have categorical variables, \(x = \set{\text{smoker}, \text{nonsmoker}}\) and \(y = \set{\text{lung cancer}, \text{no lung cancer}}\). How do we determine the relationship between these variables, given a dataset of samples?
We have been doing descriptive statistics so far  describing statistical phenomena that we can see with numerical and graphical techniques to show features of interest (e.g., data mining, data analysis).
Now we will focus on statistical inference  predicting statistical phenomena that we can't see, and generalizing conclusions from studies to the whole process/population (i.e., machine learning). It's a form of inductive reasoning.
The types of problems we'll look at in this course are:
We use our numerical and graphical summaries to choose a statistical models (mathematical models incorporating probability) to fit the data. For example, we can use statistical models to fit the exchange rate between CAD and Euro, or when earthquakes/natural disasters are likely to happen. Statistical models can even be used to simulate experiments using computers.
A random variable is a variable that represents a variate of a randomly selected unit of a population/process.
Statistical models are chosen based on background knowledge about populations (for example, we might have traffic accident data, which can be a Poisson process due to the way the process works), past experience (human height data is usually normally distributed), mathematical convenience (using easytounderstand and easytouse models where possible, like Guassian), and datasets that the models will be tested against (for example, we might be testing against normally distributed data).
In STAT230 we covered a lot of different families of statistical models, like:
The PDF of a random variable is \(f(y; \theta) = \frac{\dee}{\dee y} P(Y \le y) = \frac{\dee}{\dee y} F(y)\) where \(y \in \mathrm(range)(Y)\)  we want to show that the model is dependent on the value of the statistical model family's parameters.
Tutorial test 1 was today. Start working on chapter 2 end of chapter problems.
The tutorial test had a few questions about parameter estimation  estimating the parameters for a statistical model based on a sample of a population/process, and quantoify how good those estimates are.
For example, we can estimate \(\mu\) in a normal distribution as the sample mean, or \(\sigma^2\) as the sample variance. We could also use the sample proportion to estimate \(p\) in a Binomial distribution. We figured this out intuitively, but there are formal ways to estimate parameters too. In this course, we'll look at the Method of Maximum Likelihood (MLE), the most often used method. Another common one is the Bayesian Estimation Method.
A point estimate of a parameter \(\theta\) is an estimate \(\hat \theta\) that is the result of a function (called a point estimator) of observed data \(y_1, \ldots, y_n\). For example, the sample mean \(\overline y\) is a point estimate for the parameter \(\mu\) in a Guassian distribution.
Suppose we have a coin that has probability \(\theta\) of being heads. How do we estimate \(\theta\)? Intiutively, to get the best guess/estimate we can from the available data, we could use a Binomial model, flip the coin a lot of times, and take the sample proportion to estimate the model's \(p\) parameter. We could also keep flipping the coin until we got some number of failures, and use a negative binomial model.
The best guess is not the same thing as a good/reasonable guess  a good/reasonable guess is just one that makes the result we got relatively probable. A bad guess would make the result we got unlikely, like guessing that our coin was fair if we only got 1/100 heads (it's highly improbably that the coin has 50% heads if we observe that).
The best guess of a parameter is the one that is most likely to give us the results we did  the maximum likelihood estimate \(\hat \theta\) (note that this is the same as the point estimate). For example, if we flip a coin 100 times and get 45 heads, assuming a binomial model, the value of parameter \(p\) that is most likely to give us this result is 0.45.
In other words, if we graph \(P(Y = 45)\) where \(Y ~ \mathrm{Binomial}(100, p)\) over all possible \(p\), the graph has a peak at \(p = 0.45\). Any procedure that finds the parameter that gives us the maximum likelihood estimate is called a Method of Maximum Likelihood. Basically, we decide how plausible each value of \(\theta\) is by looking at how probably it makes our observations.
Note that \(Y\) is a random variable representing the potential data used to estimate \(\theta\), while \(y\) is the actual, observed data. Also, the set of possible values of \(\theta\) is often called \(\Omega\), the parameter space.
The likelihood function for \(\theta\) in a statistical model is defined as \(L(\theta; y) = P(Y = y; \theta) = P(Y = y; \theta)\) (the probability of observing data \(y\) as a function of \(\theta\)), sometimes shortened as \(L(\theta)\). The maximum likelihood estimate is the value of \(\theta\) that maximizes the likelihood function. Usually, we do this by differentiating the likelihood function with respect to \(\theta\), then setting that derivative to 0 and solving for \(\theta\) to find the critical points, then checking to see if they're the global maximum.
For example, the likelihood function for \(\theta\) in a binomial model is \(L(\theta; y) = P(Y = y; \theta) = {n \choose y} \theta^y (1  \theta)^{n  y}\)  we just substituted \(y\) in place of \(k\).
The value of the likelihood function doesn't actually matter, only how large it is relative to other values of the likelihood function. That means we can drop constant factors and terms in the likelihood function, since the maximum would still occur at the same \(\theta\) value. For example, we can just as well use \(L(\theta; y) = \theta^y (1  \theta)^{n  y}\).
The likelihood function is the probability of observing the data \(y\) as a function of the unknown constant \(\theta\), scaled by an arbitrary positive constant \(k\).
The relative likelihood function is defined as \(R(\theta; y) = \frac{L(\theta; y)}{L(\hat \theta)}\)  the likelihood function rescaled to be between 0 and 1. This makes comparing likelohood functions easier.
The binomial relative likelihood function is \(\frac{\theta^y (1  \theta)^{n  y}}{\hat \theta (1  \hat \theta)^{n  y}}\) where \(\hat \theta = \frac y n\).
What is the Y axis of the relative likelihood function? It's the likelihood of the X axis, not the probability of anything. Likewise, \(\theta\) isn't a probability either, just an unknown constant. For example, a value of \(\theta\) that makes \(R(\theta) = 0.6\) means that \(\theta\) is 0.6 times as likely as the maximum likelihood estimate.
A likelihood is different from a probability  likelihoods are the frequency of certain hypotheses out of all hypotheses, while probabilities are the frequency of certain results out of all results. Hypotheses are statements like "the unknown parameter \(\theta\) is 0.3", and results are something like "we observed 300 of 1000 trials were successful".
Likewise, a probability function is what we use when we have a parameter value and want to describe what samples will look like, while the likelihood function is what we use when we have a sample and want to describe what the parameter value looked like.
The log likelihood function is just the log of the likelihood function, \(l(\theta) = \ln L(\theta)\). Clearly, it has its maximum at the same point, even though the shape is different.
We like the log likelihood function because it's often easier to differentiate, which makes it easier to find the maximum likelihood estimate. Since likelihood functions are often products of lots of factors, using the product rule over and over again quickly gets messy. The log likelihood function can often be written as multiple terms using the \(\ln a + \ln b = \ln(ab)\) identity, after which we can differentiate individual terms really easily using the sum rule.
For example, the log likelihood function is \(l(\theta; y) = \ln \left(\theta^y (1  \theta)^{n  y}\right) = y \ln \theta + (n  y) \ln (1  \theta)\). Differentiating this is a piece of cake, and we get \(\frac{\dee}{\dee \theta} l(\theta) = \frac y \theta  \frac{n  y}{1  \theta} = \frac{y  \theta n}{\theta(1  \theta)}\). Solving \(\frac{y  \theta n}{\theta(1  \theta)} = 0\), we get \(\theta = \frac y n\), as expected.
Suppose we have two independent data sets \(y, z\) for two independent random variables \(Y, Z\) that share a parameter \(\theta\). Clearly, \(P(Y = y, Z = z; \theta) = P(Y = y; Z = z; \theta)\), so the combined likelihood is \(L(\theta; y) L(\theta; z)\).
Suppose \(y_1, \ldots, y_n\) is the number of events that occurred at day 1 to \(n\), respectively, and a Poisson distribution is appropriate for this dataset. What's the likelihood function for the Poisson distribution \(Y \sim \mathrm{Poisson}(\theta)\), where \(\theta\) is the average rate of events?
Intuitively, we know that the average rate of accidents is \(\overline y\). Why is this?
Here, the likelihood function is \(L(\theta; y_1, \ldots, y_n)P(\text{observing the data } y_1, \ldots, y_n; \theta)\)  the probability of observing the events we did for any given average event rate.
Let \(Y_i\) be the number of events recorded on day \(i\). Then \(L(\theta; y_1, \ldots, y_n) = P(Y_1 = y_1, \ldots, Y_n = y_n; \theta)\). Since each event occurring is independent (one of the assumptions necessary to use the Poisson distribution), \(P(Y_1 = y_1, \ldots, Y_n = y_n; \theta) = P(Y_1 = y_1; \theta) \ldots P(Y_n = y_n; \theta) = \prod_{i = 1}^n \frac{\theta^{y_i} e^{\theta}}{y_i!} = \frac{1}{\prod_{i = 1}^n y_i!} \theta^{\sum_{i = 1}^n y_i} e^{n\theta}\).
To simplify the equation, we can use the fact that \(n\overline y = \sum_{i = 1}^n y_i\). Then, dropping the constant factors and substituting \(n \overline y\) in, we get \(L(\theta; y_1, \ldots, y_n) = \theta^{n \overline y} e^{n\theta}\). So, the log likelihood function is \(l(\theta; y_1, \ldots, y_n) = n \overline y \log \theta  n \theta\).
Differentiating, \(l'(\theta) = \frac{n \overline y}{\theta}  n\), so if \(l'(\theta) = 0\), \(\theta = \overline y\).
So \(\hat \theta = \overline y\)  the sample mean. Also, the relative likelihood function is \(\frac{\theta^{n \overline y} e^{n \theta}}{{\hat \theta}^{n \overline y} e^{n \hat \theta}}\).
Recommended problem: course notes, example 2.2.4 (page 5758)
Suppose we have a randomly selected sample \(y = \tup{y_1, \ldots, y_n}\) from a random variable \(Y = \tup{Y_1, \ldots, Y_n}\). Since each \(Y_i\) is independent of the others, \(L(\theta; y) = P(Y_1 = y_1, \ldots, Y_n = y_n) = P(Y_1 = y_1) \cdots P(Y_n = y_n)\).
For discrete distributions so far we had \(L(\theta; y) = P(Y = y; \theta)\). However, this isn't suitable for continuous random variables, since \(P(Y = y; \theta)\) is always 0 (what's the probability that a dart hits a particular exact point?).
How do we construct a likelihood function? The basic idea is that whenever we're taking an observation in real life, we're actually rounding it to some number of decimal places. As a result, we can treat those observations as if they fall into lots of tiny intervals.
Basically, we can replace \(P(Y = y)\) with the probability density function (PDF) \(f(y; \theta)\). So, \(L(\theta; y) = \prod_{i = 1}^n f(y_i; \theta)\).
What's the likelihood function for an exponential distribution \(Y \sim \mathrm{Exponential}(\theta)\) for a dataset \(y_1, \ldots, y_n\), where \(\theta\) is the mean rate of events? Note that the PDF is \(f(y; \theta) = \theta e^{\frac y \theta}\).
Clearly, \(L(\theta; y) = \frac{e^{\frac 1 \theta \sum_{i = 0}^n y_i}}{\theta} = \frac{e^{\frac{n \overline y}{\theta}}}{\theta}\). Dropping the constant factors, we get \(e^{n} e^{\frac{n\overline y}{\theta}}\). So \(\hat \theta = \overline y\), the sample mean again.
What's the likelihood function for a Gaussian distribution \(Y \sim \mathrm{Guassian}(\mu, \sigma^2)\) for a dataset \(y_1, \ldots, y_n\), where \(\mu, \sigma^2\) are the mean and variance? Here, we have two parameters for our likelihood function, so we'll need to use multivariate calculus to maximize it.
Details are in the course notes, but \(\hat\mu = \overline y\) (the sample mean), and \(\sigma^2 = \frac 1 n \sum_{i = 1}^n (y_i  \overline y)^2\) (sort of like the sample variance, but with \(n\) rather than \(n  1\) for the denominator).
Also, if we know one of the parameters, we can simply use a partial differential for the likelihood function to maximize the likelihood of the unknown parameter.
Midterm next tuesday. Midterm covers everything up to but not including Monday's lecture. Do the practice problems at the end of chapter 2.
The invariance property says that if \(\hat \theta\) is the MLE of \(\theta\), then \(g(\hat \theta)\) is the MLE of \(g(\theta)\). In other words, to find the maximum likelihood of a function, we just need to find the maximum likelihood of its parameter.
For example, the variance of the binomial distribution is \(\sigma^2 = n\theta(1  \theta)\). Therefore, the MLE of the variance is \(n \hat \theta(1  \hat \theta)\)
Aside, if \(Y_1, \ldots, Y_n\) are independent random variables with \(E[Y_i] = \mu\) and \(\mathrm{Var}(Y_i) = \sigma^2\). Then \(E[\overline Y] = \mu\) and \(\mathrm{Var}(\overline Y) = \frac{\sigma^2}{n}\).
When we solve statistical problems, we usually start by fitting a model to the data. After we fit a model, however, we should always check that the model actually represents the distribution well. Some ways we can do this are: superimposing a PDF onto the relative frequency histogram and visually checking similarity, superimposing an empirical CDF on the theoretical CDF and visually checking similarity, checking the QQplot for straight lines, and comparing observed frequencies with expected frequencies predicted by the model.
The last one might warrant some explanation. One way to check model fit is to compare how often we see some samples to how often the model says we should see them. For example, for a Poisson model we would first fit it to the dataset using the sample mean as \(\theta\). Then, we can calculate the theoretical frequency for an interval \(nP(y_1 \le Y \le y_2)\) predicted by the Poisson model, and then compare them with the actual number of times we observe values between \(y_1\) and \(y_2\). We usually choose around 1015 intervals for \(y_1\) and \(y_2\) such that each one contains at least one sample.
The PPDAC is an algorithm for designing statistical studies. It consists of the following steps:
For example, a real Vitamin D study has the following PPDAClike structure:
;wip: learn about chi squared and t squared distributions
Do chapter 3 end of chapter problems. Start reading chapter 4. Start doing assignment 2, and make sure to read the instructions about the ID carefully in the assignment template.
On assignments and exams, we'll often be given a plan, and asked to critique strengths and weaknesses, sources of error, identify target population, study population, variates of interest, units, etc.
For example, if we were conducting a study about financial habits of university students in Ontario, it would be a study error to only study UW students (because of the coop program would affect student finances), and it would be a sample error to only sample UW students who were attending to a particular football game (because people who go to football games might systematically have different financial situations).
When running the study, we want to take as random a sample as possible. We might try sending out an email to everyone with a UW email, but then the students themselves choose whether they're in the sample, a possibly large source of sample error. We could try posting in the UW Facebook group, but then only people who use Facebook would see it. We might ask professors to put them on final exams, but some courses don't have final exams, for example. Instead, we might go to the university registrar, randomly select students by student ID, and then ask them to participate in a study with enough monetary incentive that they are mostly convinced to participate.
See section 3.3 of the course notes for a case study.
When we have a dataset and a Gaussian model is a good fit for the data, we usually estimate the parameter \(\mu\) using the sample mean, and the parameter \(\sigma\) with the sample standard deviation (even though it might be slightly off for a small population, since we divide by \(n  1\) rather than \(n\)).
Note that \(\hat \theta\) is really only an estimate  we would expect that it approaches the true value of \(\theta\) for large sample sizes. How close would \(\hat \theta\) be to \(\theta\)? How certain is our estimate? To quantify this, we can use sampling distributions and point estimators.
Suppose for our study population we have observed data \(y_1, \ldots, y_n\), and the attribute of interest is fully represented by the parameter \(\theta\).
A point estimate for that parameter \(\theta\) is a function \(\hat \theta = g(y_1, \ldots, y_n)\), used to estimate the unknown parameter \(\theta\). For example, a point estimate for \(\theta\) in a Poisson distribution is \(\hat \theta = \overline y\), and a point estimate for \(\theta\) in a Binomial distribution is \(\hat \theta = \frac {\sum y_i} n\).
Since the value of \(\hat \theta\) is potentially going to be different for each possible sample we take, \(\hat \theta\) has a distribution. In fact, we can represent the potential samples \(y_1, \ldots, y_n\) we can draw with \(Y_1, \ldots, Y_n\)  each unit in our sample can be thought of as being drawn from a given distribution.
We therefore associate the point estimate function with the point estimator function, \(\widetilde \theta = g(Y_1, \ldots, Y_n)\). Note that \(\widetilde \theta\) is now a random variable, rather than a simple value (this is also not standard notation, by the way). For example, a point estimator for \(\theta\) in a Poisson distribution is \(\widetilde \theta = \overline Y\), and a point estimator for \(\theta\) in a Binomial distribution is \(\hat \theta = \frac {\sum Y_i} n\).
Recall that a random variable is simply a function. \(\widetilde \theta\) is therefore a rule that tells us how to obtain an estimate of \(\theta\) for an abstract sample \(Y_1, \ldots, Y_n\), while \(\hat \theta\) is the value obtained using this rule for a concrete, observed sample \(y_1, \ldots, y_n\).
The method of maximum likelihood is one general method for getting point estimates/point estimators. Others include the method of moments and bayesian estimation.
The sampling distribution is the distribution of a point estimator \(\widetilde \theta\)  it must have a probability function if it's discrete, and a probability density function if it's continuous.
How do we figure out what the sampling distribution is? Technically, we could figure out the sampling distribution with brute force, by enumerating every possible sample and getting the point estimate for each of them. However, this is generally not feasible. Instead, we can use the central limit theorem.
Recall the central limit theorem  if we average together enough iid (independent and identically distributed) random variables \(Y_1, \ldots, Y_n\), they start to resemble a Gaussian distribution where the mean is about \(E(Y_1) = \ldots = E(Y_n)\) and the variance is \(\frac{\mathrm{Var}(Y_1)}{\sqrt{n}} = \ldots = \frac{\mathrm{Var}(Y_n)}{\sqrt{n}}\). "Large enough" is somewhat ill defined, and depends on the shape of the distributions, but symmetric distributions are faster.
If \(X \sim \mathrm{Poisson}(\mu_1), Y \sim \mathrm{Poisson}(\mu_2)\), then \(X + Y \sim \mathrm{Poisson}(\mu_1 + \mu_2)\). When \(\mu \ge 5\), a Poisson random variable will start to resemble a \(\mathrm{Guassian}(\mu, \mu)\) distribution.
If \(X \sim \mathrm{Binomial}(n_1, \theta), Y \sim \mathrm{Binomial}(n_2, \theta)\), then \(X + Y \sim \mathrm{Binomial}(n_1 + n_2, \theta)\).
If \(Y_i \sim \mathrm{Gaussian}(\mu_i, \sigma_i^2), 1 \le i \le n\), then \(\sum a_i Y_i \sim \mathrm{Gaussian}(\sum a_i \mu_i, \sum a_i^2 \sigma_i^2), a_i \in \mb{R}\). In other words, any linear combination of Gaussian random variables also forms a Gaussian distribution. From this, we know that if \(\mu = \mu_1 = \ldots = \mu_n\) and \(\sigma^2 = \sigma_1^2 = \ldots = \sigma_n^2\) (all of the normal random variables are iid), then \(\overline Y \sim \mathrm{Gaussian}(\mu, \frac{\sigma^2}{n})\).
When we take a sample \(Y_1, \ldots, Y_n\) from our population, each entry in the sample can be thought of as iid. Using this observation, we can approximate the mean of different distributions with:
For more complicated cases, we can simulate experiments using software like R.
Tutorial test 2 is coming up, will include PPDAC question and estimators.
A few classes ago we conducted the diamond experiment to estimate an unknown attribute of the study population, the mean of the diamond values, Suppose we observe data \(y_1, \ldots, y_n\), and that we have a probability model with unknown parameter \(\theta\). Let \(\hat \theta = g(y_1, \ldots, y_n) = \overline y\) be a point estimate of \(\theta\). How good is \(\hat \theta\) as an estimate of \(\theta\)? What is the uncertainty in the estimate? To help us answer these questions, we can use interval estimation rather than point estimation.
Recall that the sampling distribution is the distribution of a point estimator \(\widetilde \theta\). This is essentially the distribution of estimates of our unknown parameter \(\theta\), over all possible ways we could sample from the study population. We can approximate this by taking lots and lots of samples, estimating \(\theta\) based on each sample, and form the distribution out of all the estimates we get.
The sampling distribution of a point estimator when we're trying to estimate the mean has \(E(\overline Y) = \mu\) and \(\mathrm{Var}(\overline Y) = \frac{\sigma^2}{n}\).
Given a sampling distribution, one common question we use to quantify the uncertainty is "what proportion of our point estimates within \(d\) units of the true value of the unknown parameter?". We can easily answer this using histograms of sampling distributions. Factors that affect this uncertainty are the sample size, variance, and the shape of the sampling distribution.
In a real empirical study, we only get one sample  we can't just repeatedly resample. How do we quantify uncertainty only from one sample?
Suppose we have a dataset \(y_1, \ldots, y_n\) sampled from our study population with an assumed model \(Y_i \sim \mathrm{Gaussian}(\mu, 0.5), 1 \le i \le n\) (each number in our sample is in a Gaussian distribution where \(\sigma = 0.5\)). Suppose we're using \(\widetilde \mu = \overline Y = \frac{\sum Y_i}{n}\) to estimate \(\mu\) using our dataset. Clearly, we don't need to simulate the sampling distribution, because we know it's \(\widetilde \mu = \mathrm{Gaussian}(\mu, \frac{\sigma}{\sqrt{n}})\). Now, how often is this estimate \(\hat \mu\) within 0.1 of the true value of \(\mu\)?
In other words, we want \(P(\abs{\overline Y  \mu} \le 0.1)\) given our distribution \(\mathrm{Gaussian}(\mu, \frac{0.5}{\sqrt{n}})\), but we don't know \(\mu\). However, if we divide both sides by \(\sigma\), we get \(P(\abs{\overline Y  \mu} \le 0.1) = P(\frac{\abs{\overline Y  \mu}}{\frac{0.5}{\sqrt{n}}} \le \frac{0.1}{\frac{0.5}{\sqrt{n}}})\). Note that this is now a normalized Guassian distribution, so we have \(P(\abs{Z} \le \frac{0.1}{\frac{0.5}{\sqrt{n}}}) = P(\frac{\sqrt{n}}{5} \le Z \le \frac{\sqrt{n}}{5})\), which we can then compute using the Gaussian CDF table.
Practice problem about identifying the PPDAC steps in a news article about a study. The example is about an observational study (researchers didn't attempt to control any variates), about the relationship about women who attend science fairs/competitions at a young age and whether they go into STEM careers later on, trying to establish whether there's a causative relationship. Data was collected by an inclass survey, though we don't know if students were required to complete it.
The explanatory variate here would be whether each student is participating in science fairs/competitions, a categorical variate. The response variate is whether each student ended up going into a STEM career, and is also a categorical variate. The analysis concluded that women who participated in those things were 2.7 times more likely to go into STEM careers than women who did not  this is a relative risk numerical summary.
The sample is a subset of the study population, but the study population is not necessarily a subset of the target population.
To indicate uncertainty in an estimate, we define an interval estimate of \(\theta\) given observed data \(y\) to be \([L(y), U(y)]\), where \(L\) and \(U\) are generic interval bounding functions.
For example, the interval estimate \([\overline y  2 \frac{\sigma}{\sqrt n}, \overline y + 2 \frac{\sigma}{\sqrt n}]\) summarizes the uncertainty in the point estimate \(\hat \mu\). As the sample size increases, the confidence interval shrinks, converging on \(\overline y\).
The \(100p\) percent likelihood interval is defined as \(\set{\theta : R(\theta; y) \ge p}\), where \(\theta\) is the unknown parameter, \(R(\theta)\) is the relative likelihood function, and \(0 \le p \le 1\). Note that this isn't necessarily a single interval unless \(R(\theta; y)\) is unimodal, but all of the likelihood functions we'll look at in this course will be unimodal.
The log relative likelihood function is defined as \(r(\theta; y) = \ln R(\theta; y)\), and we use it because it's often easier to compute \(r(\theta; y)\) than \(R(\theta; y)\). The log relative likelihood function often looks pretty quadratic, while the relative likelihood function often looks like a bell curve. We sometimes write the \(100p\) percent likelihood interval as \(\set{\theta : r(\theta; y) \ge \ln p}\).
It's important to note that the likelihood interval is always a function of the observed data. The likelihood interval is an example of an interval estimate.
The wider the likelihood interval, the more uncertainty about our estimate, and the less we know about the value of the unknown parameter. Values in the 10% likelihood interval are generally considered plausible. Values inside the 50% likelihood interval are generally considered very plausible. Values outside the 1% likelihood interval are generally considered very implausible.
We usually find the likelihood interval either by solving the inequality, or by drawing a horizontal line at \(y = p\) on the graph of the relative likelihood function and reading off the Xaxis values that are above the line.
How often does the likelihood interval contain the true parameter value? An interval estimator can answer this question, much like point estimators did for point estimates.
We again represent our potential dataset as random variables \(Y = Y_1, \ldots, Y_n\) (as opposed to the dataset \(y = y_1, \ldots, y_n\)). Then, we have the interval estimator \([L(Y), U(Y)]\) where ;wip: likelihood estimator slides
To determine how good the interval estimator is we want to compute \(P(L(Y) \le \theta \le U(Y))\). This gives us the concept of the confidence interval  what's the narrowest possible interval \([L(Y), U(Y)]\) that has a \(100p\) percent chance of containing the true value of \(\theta\)?
The \(100p\) percent confidence interval is the interval estimate \([L(y), U(y)]\) such that \(P(L(y) \le \theta \le U(y)) = p\), where \(\theta\) is the unknown parameter and \(0 \le p \le 1\). The center of the confidence interval is always the point estimate for the mean (i.e., the sample mean), so the confidence interval is just the point estimate plus or minus the margin of error. Also, \(p\) is called the confidence coefficient.
It's not valid to say that a 99% confidence interval means that we'd expect the true value to have a 99% probability of being in the interval, because the true value of \(\theta\) is just a constant, so doesn't have a distribution or probability of being in there  it's either in the interval or not in the interval, and we're only doing the study once. However, we can say that a 99% confidence interval means that we would be 99% confident that the true value is in the interval  if we did the experiment 100 times, we'd expect the interval we got to contain the true value 99 times.
Turns out the confidence interval can be pretty easily calculated. We calculate the sample mean \(\overline y\) and sample standard deviation \(s\), and then choose a value. Then, the confidence interval is \([\overline y  z^* \frac{s}{\sqrt n}, \overline y + z^* \frac{s}{\sqrt n}]\)
Suppose \(Y_1, \ldots, Y_n\) is a random sample of a \(\mathrm{Gaussian}(\mu, 1)\) distribution. Then \(E(Y_i) = \mu\) and \(\mathrm{Var}(Y_i) = 1\).
Consider the interval \([\overline Y  \frac{1.96}{\sqrt n}, \overline Y + \frac{1.96}{\sqrt n}]\). What's the probability that \(\mu\) is in this interval, or \(P(\overline Y  \frac{1.96}{\sqrt n} \le \mu \le \overline Y + \frac{1.96}{\sqrt n})\)?
This is a somewhat unusual form for a probability calculation, because the random variable appears on the range bounds and the inner value \(\mu\) is a constant. To solve this using the techniques we learned in STAT230, we want to rewrite this so the random variable is in the middle.
Subtracting \(\overline Y\) and multiplying by \(\sqrt{n}\), we get \(P(1.96 \le \sqrt{n} (\mu  \overline Y) \le 1.96)\). We can rewrite this as \(P(1.96 \le \frac{\overline Y  \mu}{\frac 1 \sqrt{n}} \le 1.96)\).
Clearly, \(\overline Y \sim \mathrm{Gaussian}(\mu, \sigma = \frac{1}{\sqrt n}})\), because it's the average of \(n\) samples. Therefore, \(\frac{\overline Y  \mu}{\frac 1 \sqrt{n}} \sim \mathrm{Gaussian}(0, 1)\), so we can find the value of \(P(1.96 \le \frac{\overline Y  \mu}{\frac 1 \sqrt{n}} \le 1.96)\) using the Gaussian CDF table.
Clearly, the width of the table is \(2 \frac{1.96}{\sqrt n}\), which decreases as \(n\) increases.
If we had a sample of size 16 with mean 10.4, we would have a 95% confidence interval \([10.4  \frac{1.96}{\sqrt{16}}, 10.4 + \frac{1.96}{\sqrt{16}}]\). This tells us that we are 95% confident that \(10.4  \frac{1.96}{\sqrt{16}} \le \mu \le 10.4 + \frac{1.96}{\sqrt{16}}\). Note that this is not the same thing as \(P(10.4  \frac{1.96}{\sqrt{16}} \le \mu \le 10.4 + \frac{1.96}{\sqrt{16}}) = 0.95\)  all of the values in that statement are constants, so the probability is either 0 or 1, we just don't know which it is. If we repeated the experiment many times, the range would include the true value \(100p\) percent of the time (though we very rarely get the chance to repeat our experiments).
A pivotal quantity \(Q(Y; \theta)\) is a function of a random sample \(Y\) and unknown parameter \(\theta\) such that the distribution of \(Q(Y; \theta)\) is completely known. That makes \(P(a \le Q(Y; \theta) \le b)\) depends only on \(a\) and \(b\). The pivotal quantity must be a function of the data and unknown parameter, so \(\overline Y\) is not a pivotal quantity while \(\overline Y  \mu\) is.
In this course we won't need to find pivotal quantities on our own, but we should be able to show why a given function is a pivotal quantity. We also need to be able to construct a confidence interval using a pivotal quantity.
To use a pivotal quantity to find a \(100p\) percent confidence interval, we determine values \(a, b\) such that \(P(a \le Q(Y; \theta) \le b) = p\). Then, we rewrite \(a \le Q(Y; \theta) \le b\) as \(L(Y) \le \theta \le U(Y)\). Then, we use the fact that \(p = P(a \le Q(Y; \theta) \le b) = P(L(Y) \le \theta \le U(Y))\) to solve for \(L(Y), U(Y)\). The interval \([L(Y), U(Y)]\) is then the \(100p\) percent confidence interval.
For \(Y \sim \mathrm{Gaussian}(\mu, \sigma)\) where \(\sigma\) is known, we use the normal table to find a value such that \(P(a \le Z \le a) = p\) (equivalently, \(P(Z \le a) = \frac{1 + p}{2}\), which should be easier to look up in the normal table), and then the \(100p\) percent confidence interval for samples is is \([\overline y  a \frac{\sigma}{\sqrt n}, \overline y + a \frac{\sigma}{\sqrt n}]\). This is also the case for other distributions if the sample is large enough, because of the central limit theorem.
Common confidence intervals we care about are 90%, 95%, and 99%, with values of \(a\) equal to 1.645, 1.960, and 2.576 respectively.
Suppose we have an empirical study of hand/leg dimensions of students in class. The target population might be the set of students taking STAT231, while the sample population might be the set of students that showed up to the class that the study was taking place in. This is an observational study, since even though the experimenter asked us to do something, it was not something that would affect our variate of interest.
Tutorial test 2 on wednesday. Check information on LEARN.
Recall the definition of a pivotal quantity: a function of the data and the unknown parameter such that the distribution is fully known. Recall that to turn a pivotal quantity into a confidence intervals, we find the bounds of the pivotal quantity, and then transform the expression until we get the form \(P(L(Y) \le \theta \le U(Y)) = p\), or the confidence interval \([L(Y), U(Y)]\).
Pivotal quantities are often hard to find analytically. However, we often use approximate pivotal quantities that apply for values as the sample size gets large, due to the central limit theorem.
Recall that if \(Y \sim \mathrm{Binomial}(n; \theta)\), then \(E(Y) = \theta\) and \(\mathrm{Var}(Y) = \frac 1 n \theta(1  \theta)\). By the central limit theorem, the approximate sampling distribution of \(\widetilde \theta = \frac Y n\) is \(\frac{\widetilde \theta  \theta}{\sqrt{\frac 1 n \theta(1  \theta)}}\) for a large enough value of \(n\).
However, this formula has too many \(\theta\)  the function is too complicated to use as a pivotal quantity. Instead, we can replace some of our \(\theta\) isntances with \(\widetilde \theta\), to get \(Q_n = \frac{\widetilde \theta  \theta}{\sqrt{\frac 1 n \widetilde \theta(1  \widetilde \theta)}}\). This is an approximate pivotal quantity for the binomial distribution.
We can use this to construct approximate confidence intervals, using the usual way of converting a pivotal quantity into a confidence interval. Given a large \(n\), a \(100p\) percent approximate confidence interval for \(\theta\) is \(\theta \pm a \sqrt{\frac 1 n \hat \theta (1  \hat \theta)}\) where \(a\) is defined the same way as for the Gaussian confidence interval above.
;wip: confidence interval formula for Poisson
We now have two types of interval estimates  \(100p\) percent likelihood intervals, and \(100p\) percent confidence intervals. There's actually an inportant connection between the two types of interval estimates, which we'll look at later.
Researchers use confidence intervals to choose how large to make their samples. They compute the sample size their studies need based on a worstcase confidence interval width (and available resources like budget and manpower).
Most interval estimates have the form \(\text{point estimate} \pm \text{uncertainty}\).
Likelihood intervals are just a different way to write approximate confidence intervals. To show this, we use the \(\Chi^2(k = 1)\) distribution, which looks like a downward upconcave slope.
The maximum likelihood estimator of \(\theta\) is \(A = 2 \ln \frac{L(\theta; Y)}{L(\widetilde \theta; Y)}\). \(A\) is a random variable that depends on the random data \(Y = Y_1, \ldots, Y_n\), and is also called the likelihood ratio statistic (likelihood ratio random variable).
We won't show why this works in this class, but it turns out that with a large enough \(n\), \(A\) roughly has a \(\Chi^2(1)\). In other words, with a large enough sample, we can use the Chisquared distribution to approximate the likelihood ratio statistic. That means \(A\) is approximately a pivotal quantity for \(\theta\).
The "large enough" value of \(n\) depends on the distributions of the individual samples, just like the central limit theorem  30 is usually a good value for this course. The above statement is actually quite similar to the central limit theorem, and the likelihood ratio statistic is somewhat similar to the relative likelihood function.
We can then use the likelihood ratio statistic as an approximate pivotal quantity to create \(100p\) percent confidence intervals. First, we find a \(c\) such that \(p = P(W \le c)\) where \(W \sim \Chi^2(1)\). Now, we note that \(P(W \le c) \approxeq P(2 \ln \frac{L(\theta; Y)}{L(\widetilde \theta; Y)} \le c)\) for large enough \(n\).
Therefore, the approximate \(100p\) percent confidence interval is \(\set{\theta: 2 \ln \frac{L(\theta; y)}{L(\hat \theta; y)} \le c}\), or \(\set{\theta: 2 \ln R(\theta; y) \le c}\), or \(\set{\theta: R(\theta; y) \ge e^{\frac c 2}}\). But, note that this is just a likelihood interval!
Note that if \(W \sim \Chi^2(1)\) and \(Z \sim \mathrm{Gaussian}(0, 1)\), then \(P(W \le c) = 2 P(Z \le \sqrt c)  1\)  we can convert between Chisquared and Guassian CDF values exactly. This is useful for when our Chisquared tables don't go up to our desired values.
For example, since \(0.95 = 2P(Z \le 1.96)  1\), \(0.95 = P(W \le 1.96^2)\) and an approximate confidence interval for \(\theta\) is \(\set{\theta: R(\theta) \ge e^{\frac{1.96^2}{2}}} = \set{\theta: R(\theta) \ge 0.147}\).
Therefore, the 14.7% likelihood interval is the same thing as an approximate 95% confidence interval (for large enough sample sizes)! In the same way, the 4% likelihood interval is a 99% confidence interval, a 10% likelihood interval is a 97% confidence interval, and a 26% likelihood interval is a 90% confidence interval.
Note that as \(p\) increases a likelihood interval gets narrower, while a confidence interval gets wider.
So to convert a \(100p\) percent likelihood interval into a confidence interval (given a large enough sample size), we let \(c = 2 \ln p\), and then look up the value \(k\) such that \(k = P(W \le c)\) in the Chisquared CDF table, or \(k = 2P(Z \le \sqrt c)  1\) in the Gaussian CDF table.
Consider the binomial distribution. We can now find the 95% approximate confidence interval with the 14.7% likelihood interval, or using the other approach \(\overline y \pm \hat \theta \sqrt{\frac{\hat \theta (1  \hat \theta)}{n}}\). The former is based on the Chisquared approximation, while the latter is based on the central limit theorem approximation.
The former tends to be a better approximation for smaller sample sizes, but is often harder to calculate as well. In fact, likelihood interval estimates are at least as good as pivotalquantitybased estimates, in general.
We already looked at the confidence interval for the mean of a Gaussian distribution when \(\sigma\) is known, but what about when we don't know \(\sigma\)?
Clearly, \(\overline Y\) is a point estimator for \(\mu\), and \(\mathrm{Var}(Y)\) is a point estimator for \(\sigma^2\) (the sample variance). Note that the point estimator for \(\sigma\) is NOT the MLE, \(\frac{}{}\). We use the sample variance instead of the MLE because \(E(\mathrm{Var}(Y)) = \sigma^2\)  it's an unbiased estimator. ;wip: it's not Var(Y), copy the real one and compare to MLE, write in terms of estimator \(S\)
The confidence interval for \(\mu\) is still \(\overline y \pm a \frac{\sigma}{\sqrt n}\), but we have two unknowns now, which makes it hard to make a pivotal quantity. Instead, we replace \(\sigma\) with the estimator \(S\) to get the pivotal quantity \(\frac{\overline Y  \mu}{\frac{S}{\sqrt n}}\). It turns out that this pivotal quantity exactly has the \(\mathrm{studentt}(n  1)\) distribution. The studentt distribution has \(n  1\) degrees of freedom rather than \(n\) because \(\sum (Y  \overline Y) = 0\)  when we estimated the mean, we essentially locked away one degree of freedom.
This is a pivotal quantity for \(\mu\) because \(S\) is a random variable that's a function of the data, so the whole thing is a function of the data and the unknown parameter \(\mu\) whose distribution is completely known.
Therefore, we have the \(100p\) confidence interval for \(\mu\) \(p = P(a \le \frac{\overline Y  \mu}{\frac{S}{\sqrt n}} \le a)\), where \(a\) is a value such that \(P(T \le a) = \frac{1 + p}{2}\) where \(T \sim \mathrm{studentt}(n  1)\). Rearranging the pivotal quantity, we get \(p = P(\overline Y  a\frac{S}{\sqrt n} \le \mu \le \overline Y + a \frac{S}{\sqrt n})\), or a confidence interval of \([\overline y  a\frac{s}{\sqrt n}, \overline y + a\frac{s}{\sqrt n}]\). This is actually an exact confidence interval  we didn't make any distribution approximations!
To summarize, the \(100p\) percent confidence interval of a Guassian distribution where both \(\mu\) and \(\sigma\) are unknown is \(\overline y \pm a\frac{s}{\sqrt n}\) where \(a\) is a value such that \(P(T \le a) = \frac{1 + p}{2}\) and \(T \sim \mathrm{studentt}(n  1)\) and $s = $ ;wip
Midterm next week, covers everything up to and including section 5.2. Review of content before reading week.
How do we find a \(100p\) percent confidence interval for \(\sigma^2\) in a Gaussian distribution where \(\mu\) is unknown?
For a sample of size 25 drawn from a Gaussian distribution with unknown \(\mu\) and \(\sigma^2\), then \(\overline y \pm a \frac{s}{5}\) is a 95% confidence interval for \(\mu\) if \(P(T \le a) = 0.975\) and \(T \sim t(24)\).
If \(Y_1, \ldots, Y_N\) is a random normal sample and we don't know \(\mu\), then \(\frac{(n  1)S^2}{\sigma^2} = \frac{\sum (Y_i  \overline Y)}{\sigma^2} \sim \chi^2(n  1)\), and we can use this as a pivotal quantity for \(\sigma\). If we do know \(\mu\), then \(\frac{\sum (Y_i  \mu)}{\sigma^2} \sim \chi^2(n)\) instead (there's another degree of freedom that wasn't removed by the subtracting of \(\overline Y\)). We prefer \(S^2\) over the MLE for the sample variance because \(E(S^2) = \sigma^2\), while \(E(\mathrm{Var}(Y)) \ne \sigma^2\).
Note that \(\frac{\sum (Y_i  \mu)^2}{\sigma^2} \sim \chi^2(n)\). This is \(n\) rather than \(n  1\) because in the previous formula, we removed one degree of freedom by replacing \(\mu\) by \(\overline Y\).
Clearly, \(W = \frac{(n  1)S^2}{\sigma^2}\) is a pivotal quantity, so to construct a \(100p\) percent confidence interval, we want the values of \(a, b\) such that \(p = P(a \le W \le b)\). In other words, we want \(P(W \le a) = \frac{1  p}{2}\) and \(P(W \le b) = \frac{1  p}{2} + p = \frac{1 + p}{2}\). We can then look up the corresponding values of \(a, b\) in the chisquared distribution table.
Rearranging the formula, we get \(p = P(\frac{(n  1)S^2}{b} \le \sigma^2 \le \frac{(n  1)S^2}{a})\), the \(100p\) percent confidence interval for \(\sigma^2\) when \(\mu\) is unknown is \([\frac{(n  1)S^2}{b}, \frac{(n  1)S^2}{a}]\).
In north america, the criminal court results in either "the defendant is guilty" or "the defendant is not guilty". We start by assuming the defendant is innocent, and then the prosecuter tries to find enough evidence to show that the hypothesis, that the defendant is innocent, is not plausible. Meanwhile. the defence tries to show that the hypothesis is plausible. The defendant is guilty only if we can show that the hypothesis isn't plausible  we don't have to prove the hypothesis. There are two types of possible errors here: convicting an innocent person, and failing to convict a guilty person.
Statistical tests work in a way similar to these criminal trials  we start from the null hypothesis, and then design an experiment to show that our alternative hypothesis is more plausible than the null hypothesis.
The null hypothesis is the "default option", and is usually denoted \(H_0\). For example, in a drug trial, it would be "this new drug does not work", or "this previous drug is the best option for treating this disease". Likewise, if we were trying to prove someone had ESP, the null hypothesis would be "the person does not have ESP". Our experiment should be designed to falsify the null hypothesis and support our own hypothesis. The null hypothesis.
Suppose we want to tell if someone has ESP. To do this, we might design an experiment to see if someone can guess a coin flip value better than by chance. If the person guesses right 50% of the time, they probably don't have ESP. If they guess right 100%, there's probably some evidence that they do. How do we measure the strength of this evidence?
For this purpose, we need a test statistic/discrepancy measure  a function of the data \(D = g(Y)\) that measures the degree that the data \(Y\) agrees with the null hypothesis \(H_0\), where smaller values indicate more agreement between the data and the null hypothesis. In this course, we're generally given the test statistic, but we will need to be able to explain why a particular test statistic makes sense.
For example, a test statistic for the ESP experiment with 25 trials might be \(D = \abs{Y  E(Y)} = \abs{Y  12.5}\), because small values indicate that it's more likely the person is randomly guessing, while larger values indicate that it's less likely. Suppose we get a value of \(y = 15\), so \(d = 2.5\). Assuming the null hypothesis is true, what's probability of getting \(d = 2.5\)?
Clearly, \(P(D \ge 2.5 \mid H_0) = P(\abs{Y  12.5} \ge 2.5 \mid H_0)\) where \(Y \sim \mathrm{Binomial}(25, 0.5)\). So \(P(D \ge 2.5 \mid H_0) = P(Y \le 10) + P(Y \ge 15) = P(Y \le 10) + (1  P(Y \le 15)) = 0.4244\). So if the null hypothesis is true, we would expect \(D \ge 2.5\) to happen in 42% of our experiments.
Read sections 5.1 to 5.2 in the course notes.
To statistically test a hypothesis \(H\), we assume that we're testing a null hypothesis \(H_0\) using data \(Y\). Then, we set a discrepency measure \(D(Y)\) for which large values of \(D\) are less conssitent with \(H_0\), where \(d = D(Y)\) is the observed value of \(D\). We then compute the pvalue \(P(D \ge d; H_0)\).
A pvalue's meaning depends on the test statistic and the data. It's defined as \(P(D \ge d; H_0)\)  the probability of observing the discrepency measure that we did, assuming the null hypothesis is true. Essentially, for a pvalue, if we ran the experiment many times, we would expect that that percentage of the time, the experiment would be at least as unusual as the result we got, by the definition of unusual set by the discrepency statistic.
For example, the ESP experiment had the null hypothesis \(\theta = 0.5\) where \(Y \sim \mathrm{Binomial}(25, \theta)\), and the discrepency measure \(d = D(Y) = \abs{Y  n \theta} = \abs{Y  12.5}\). Suppose \(y = 15\), so \(P(D \ge d) = P(\abs{Y  12.5} \ge 15; \theta = 0.5) \approxeq\).
If the pvalue is small, then either the null hypothesis is true and we only got what we did by chance, or the null hypothesis is false. A small pvalue gives strong evidence against the null hypothesis, while a large pvalue says that there's little evidence that the null hypothesis is false.
pvalues less than or equal to 0.001 is very strong evidence of null hypothesis being false, 0.001 to 0.01 is strong evidence, 0.01 to 0.05 is evidence, 0.05 to 0.1 is some evidence, and greater values are no evidence.
For example, if we have a pvalue 0.001, then the event "observing a value as more extreme than the value observed given that the null hypothesis is true" happens only about one time out of 1000
For binomial models, \(\abs{Y  E(Y)}\) is a reasonable discrepency measure, because it's 0 when the results are the same as the null hypothesis predicts, and increases as the results differ more.
Suppose we have hypothesis \(\mu = 5\) for a Gaussian model, where . How do we calculate pvalues for that?
Recall that the pivotal quantity is \(\frac{\sqrt n (\overline Y  \mu)}{S} \sim \mathrm{studentt}(n  1)\). A sensible discrepency measure for \(\mu\) would be \(\frac{\abs{\sqrt n (\overline Y  \mu)}}{S}\), because . ;wip: why is the absolute value of the pivotal quantity a good test statistic
pvalues are often accompanied by confidence intervals  when pvalues are small for a null hypothesis, the confidence interval shows where the actual value falls.
Basically, if we use the same pivotal quantity to construct a confidence interval and a pvalue \(p\), then the hypothesis \(\theta = k\) is inside a \(100q\) percent confidence interval if and only if \(p \ge 1  q\). For example, if \(p \ge 0.05\), then \(\theta\) is inside the 95% confidence interval.
The pvalue for testing \(H_0\), a hypothesis \(\theta = k\), is greater or equal to a pvalue \(q\) if and only if \(\theta = k\) is in the \(100(1  q)\) percent confidence interval for \(\theta\), if we use the same pivotal quantity. In other words, the confidence interval is roughly the opposite of the pvalue.
There's a difference between statistical and practical significance. If you toss a fair coin millions of times, you can almost certainly find evidence against the hypothesis \(\theta = 0.5\). If we calibrated a scale and noticed that it was consistently biased 1% below the true weight, then it would have strong evidence that the scale wasn't perfectly fair, but in practice a 1% interval doesn't have any practical significance for most purposes.
Suppose we have a random Gaussian sample \(Y_1, \ldots, Y_n \sim \mathrm{Gaussian}(\mu, \sigma)\), and we're trying to test the hypothesis \(H_0\), which is \(\sigma^2 = k^2\). Recall that we have the pivotal quantity \((n  1)\frac{S^2}{\sigma^2} \sim \chi^2(n  1)\). If we assume the hypothesis, \((n  1)\frac{S^2}{\sigma^2} = (n  1)\frac{S^2}{k^2} \sim \chi^2(n  1)\).
\(S^2\) is the point estimator for \(\sigma^2\).
For normal data, we'll use the test statistic \(U = (n  1)\frac{S^2}{k^2}\) (the absolute value of the pivotal quantity). This is a reasonable test statistic because the expected value is the expected value of a \(\chi^2(n  1)\) random variable, so \(n  1\). That means \(E(\frac{S^2}{k^2}) = 1\), so the expected value of \(S^2\) is \(k^2\). In other words, if the hypothesis is true, then the value of the test statistic is close to the mean \(n  1\). If the hypothesis is false and \(\sigma^2\) is far away from \(k^2\), then the test statistic increases in value.
If \(P(U \le u) < 0.5\), then the pvalue is \(2P(U \le u)\). If \(P(U \ge u) < 0.5\), then the pvalue is \(2P(U \ge u)\), where \(U \sim \chi^2(n  1)\) (the value that is between 0 and 1 is the actual pvalue, the other value is discarded).
Likelihood ratio test statistic for a single parameter: \(R(\theta)\) is close to 1 if and only if \(2 \ln \frac{L(\theta)}{L(\hat \theta)}\) is close to 0. Likewise, \(R(\theta)\) is close to 0 if and only if \(2 \ln \frac{L(\theta)}{L(\hat \theta)}\) is large. We can therefore use the likelihood ratio as a test statistic, because it gets smaller as the value of \(\theta\) becomes more likely.
Recall that \(\Lambda(\theta) = 2 \ln \frac{L(\theta)}{L(\hat \theta)} \sim \chi^2(1)\), so we can use this to construct our pvalues with \(P(\Lambda(Y) \ge \Lambda(y); H_0)\).
So for binomial data \(Y \sim \mathrm{Binomial}(n, \theta)\), \(L(\theta) = \theta^y(1  \theta)^{n  1}\). So \(2 \ln \frac{\theta^y(1  \theta)^{n  1}}{\hat \theta^y(1  \hat \theta)^{n  1}} \sim \chi^2(1)\).
Recall that we also have the asymptotic normal statistic with \(2(1  P(Z \le ))\). Both approximations are used in this course. ;wip: slide before binomial example
Suppose we take a sample of 30 students from this term's STAT231 and their final grades in STAT231 and STAT230, and we want to look at the relationship between a student's STAT230 mark and their STAT231 mark. This is clearly bivariate data, the units are students, and the variates are final grades in STAT230 and STAT231. The study population might be those students taking STAT231 now, the sample is those 30 students we got data for, and the target population might be all STAT230/STAT231 students now and into the future. Some of the summaries we care about are sample correlation coefficient, scatter plots, and relative risk.
The explanatry variate here is the STAT230 grade, and the response variate is the STAT231 grade. To model the data, we'll need a new model we haven't looked at yet. We're mainly interested in answering questions like, "how do I predict my STAT231 grade given my STAT230 grade?".
Machine learning is just statistics. Our datasets become "training data", predictive experiments become "regressions", and fitting models becomes "training models".
Recall that the sample correlation is defined as \(S_{x, y} = \sum (x_i  \overline x)(y_i  \overline y)\). It represents the degree of the linear relationship between two variables.
The method of least squares is used to draw a line of best fit. Essentially, we're trying to minimise the sum of squares of the difference between each point and our line, on the Y axis  given a bivariate sample \(d = \tup{x_1, y_1}, \ldots, \tup{x_n, y_n}\) and a line \(\beta n + \alpha\), we want to minimise \(\sum_{\tup{x_i, y_i} \in d} \left( y_i  (\beta x_i + \alpha) \right)^2\). Note that this is not the same thing as minimizing the sum of the absolute value of the differences  we might get a different line if we do that.
Note that if you swap \(x\) and \(y\), you can potentially get a different answer  the method of least squares depends on the order of the variates. \(x_i\) should be the explanatory variate, and \(y_i\) should be the response variate  the line of best fit is used to predict how \(y_i\) should behave given \(x_i\).
To minimize this sum of squares of vertical distances, we can set its derivative with respect to \(\alpha\) to 0 and solve for \(\alpha\), do the same for \(\beta\) and solve for \(\beta\), then use the solution as critical points test them to find minima. Turns out, if we solve for this, we get \(\hat \beta = \frac{S_{x, y}}{S_{x, x}}\) and \(\hat \alpha = \overline y  \hat beta \overline x\).
Recall that the ;wip: slide relating r to alpha and beta
We now have the line of best fit \(\hat \beta n + \hat \alpha\). Clearly, this lets us predict \(y_i\) given \(x_i\), just by evaluating the line at a certain \(x\) value. But how good of a prediction does this line make? So far, we haven't defined a model, so we don't have anything to evaluate abstractly.
We need a statistical model to get an interval estimate for each prediction the line makes.
Consider the students who got exactly 75 in STAT230 as a population. If we let \(Y\) be a random variable representing the STAT231 grade of those students. Clearly, we could use a Gaussian model to represent \(Y\), so \(Y \sim G(\mu, \sigma)\). If we generalize this to all students, we might say that the mean \(\mu\) of that Gaussian distribution could be represented as a linear function \(\mu(x) = \alpha + \beta x\), and \(\sigma\) is just a constant. In other words, we have \(Y_i \sim \mathrm{Gaussian}(\alpha + \beta x_i, \sigma)\), all independently.
The line of best fit with the Gaussian model is known as simple linear regression model, because it's the simplest way to model and predict bivariate data. We're assuming that the variance is the same for each \(x_i\), so the entire model has only the unknown parameters \(\alpha, \beta, \sigma\).
Our goal for using Gaussian models is to be able to construct confidence intervals for the model predictions.
We could also assume that the variance is a function of \(x_i\), but this would be a lot harder to work with. We'll now look at graphical ways to assess whether it's reasonable to assume that the variance is constant.
Consider the parameter \(\mu(x) = \alpha + \beta x\). \(\beta\) represents the change in mean STAT231 mark given a one mark increase in STAT230 mark. \(\alpha\) represents the STAT231 mark for students that got a STAT230 mark of 0 (in this case this parameter is not really of interest, because it's not possible to get a 0 in STAT230). \(\sigma\) represents the variability in the response variate \(Y_i\) for each explanatory variable \(X_i\).
Since it's a Gaussian, we get the likelihood function \(L(\alpha, \beta; \tup{x_1, y_1}, \ldots, \tup{x_n, y_n}) = \prod \exp\left(\frac{1}{2\sigma^2} (y_i  \alpha  \beta x_i)\right)^2 = \exp\left(\frac{1}{2\sigma^2} \sum (y_i  \alpha  \beta x_i)^2\right)\), assuming \(\sigma\) is known.
Maximizing the likelihood function is equivalent to minimizing \(\sum (y_i  \alpha  \beta x_i)^2\), because maximizing \(\exp(f(x))\) is equivalent to minimizing \(f(x)\). In other words, the line of least squared error is also the maximum likelihood estimate for the line.
So the values \(\hat \beta = \frac{S_{x, y}}{S_{x, x}}, \hat \alpha = \overline y  \hat \beta \overline x\) we got above, assuming the Gaussian models are reasonable.
We have the point estimate \(\hat \beta\), but what's the distribution of the point estimator?
Clearly, \(\widetilde \beta = \frac{S_{X, Y}}{S_{X, X}} = \frac{1}{S_{x, x}} \sum (x_i  \overline x)(Y  \overline Y) = \frac{1}{S_{x, x}} \sum \frac{x  \overline x}{S_{x, x}} Y_i\).
Clearly, \(S_{X, Y} = \sum (x_i  \overline x)(Y  \overline Y) = \sum (x  \overline x) Y_i  \overline Y \sum (x_i  \overline x) = \sum (x  \overline x) Y_i  \overline Y \times 0 = \sum (x  \overline x) Y_i\).
Since \(\sum (x_i  \overline x) = \sum x_i  n\overline x = 0\), \(\sum (x_i  \overline x)(Y  \overline Y) = \sum (x  \overline x) Y_i\).
So \(\widetilde \beta = \sum \frac{x_i  \overline x}{S_{x, x}} Y_i\), which is a linear combination of Gaussians. We know that a linear combination of Guassian variables is still a Gaussian variable, where the means and variances add up. So \(\widetilde \beta \sim G(E(\widetilde \beta), \sigma(\widetilde \beta))\).
Clearly, \(E(\widetilde \beta) = \sum \frac{x_i  \overline x}{S_{x, x}} E(Y_i)\). Since \(E(Y_i) = \alpha + \beta x_i\), \(E(\widetilde \beta) = \frac{1}{S_{x, x}} \left(\alpha \sum (x_i  \overline x) + \beta \sum (x_i  \overline x) x_i\right) = \frac{1}{S_{x, x}} \left(\beta \sum (x_i  \overline x) x_i\right) = \beta\) ;wip: what's that last part
Clearly, \(\mathrm{Var}(\widetilde \beta) = \mathrm{Var}(\sum \frac{x_i  \overline x}{S_{x, x}} Y_i) = \sum \frac{(x_i  \overline x)^2}{S_{x, x}^2} \mathrm{Var}(Y_i) = \sum \frac{\sigma^2}{(S_{x, x})^2} \sum (x_i  \overline x)^2\). Since \(\sum (x_i  \overline x)^2 = S_{x, x}\), we get \(\mathrm{Var}(\widetilde \beta) = \frac{\sigma^2}{S_{x, x}}\).
So knowing \(\mu\) and\(\sigma\), we now know that \(\widetilde \beta \sim G(\beta, \frac{\sigma}{\sqrt{S_{x, x}}})\).
Since \(Y_i \sim G(\mu, \sigma)\), \(\widetilde \mu = \overline Y \sim G(\mu, \frac{\sigma}{\sqrt n})\).
We want to use \(\frac{\widetilde \beta  \beta}{\sigma / \sqrt{S_{x, x}}}\) as a pivotal quantity. However, we still don't know the distribution of \(\sigma\)  we need an estimator for \(\sigma\).
One approach is to use the MLE of the variance  \(\hat \sigma^2 = \frac{1}{n} \sum (y_i  \hat \alpha  \hat \beta x_i)\). However, this is a biased estimate. Instead, we'll use \(s_e^2 = \frac{1}{n  2} \sum(y_i  \hat \alpha  \hat \beta x_i)\)  an unbiased estimate of the variance (i.e., \(E(S_e^2) = \sigma^2\)). There's a proof for this, but it's out of scope for this course.
Now consider what happens when we replace \(\sigma\) with its unbiased estimate: \(\frac{\widetilde \beta  \beta}{s_e / \sqrt{S_{x, x}}} \sim t(n  2)\). This is a pivotal quantity for \(\beta\), which we can use to construct confidence intervals for \(\beta\) and test hypotheses like \(\beta = k\).
The \(100p\) percent confidence interval for \(\beta\) is \(\hat \beta \pm as_e \frac{1}{\sqrt{S_{x, x}}}\), where \(P(T \le a) = \frac{1 + p}{2}\). Notice that because the pivotal quantity is the studentt distribution, which is symmetric, the confidence interval is centred at \(\hat \beta\).
Usually we're interested in proving or disproving the null hypothesis: "there is no relationship between these two variables". To do this, we use the null hypothesis \(\beta = 0\)  this means that the explanatory variable doesn't affect the response variable at all. Because it's so important, the hypothesis \(\beta = 0\) is known as the "hypothesis of no relationship".
If there's a very small pvalue, we have very strong evidence against the hypothesis of no relationship, which implies that there is a relationship between the two variables.
Recall that simple linear regression has \(x\), the explanatory variate, and \(y\), the response variate, where \(x\) is assumed to be known and \(y\) is the experimental result. In the simple linear regression model, we assume that each \(y\) value has a Gaussian distribution based on constant values of \(x\).
We now have a confidence interval for \(\beta\). Using the invariance property of MLEs, we can now easily find a point estimate for \(\mu(x) = \alpha + \beta x\), as \(\hat \mu = \alpha + \hat \beta x\). However, how do we get a confidence interval?
First, we need the sampling distribution of \(\widetilde \mu(x) = \widetilde \alpha + \beta x\). Turns out, we can write this as \(\widetilde \mu(x) = \sum (\frac 1 n + (x_i  \overline x \frac{x_i  \overline x}{S_{X, X}}))Y_i\). Turns out that this is a sum of Gaussian variables, and if we do the algebra, we get \(\widetilde \mu(x) \sim G(\mu(x), \sigma \sqrt{\frac 1 n + \frac{(x  \overline x)^2}{S_{x, x}}})\).
If we know \(\sigma\), then \(\widetilde \mu(x) = \sigma + \widetilde \beta x\), and \(\hat a \pm a s_e \sqrt{\frac 1 n + \frac{(x  \overline x)^2}{S_{X, X}}}\) is a confidence interval for \(\mu(x)\). If we don't, then \(\widetilde \mu(x) = \widetilde \sigma + \widetilde \beta x\), and \(\hat a \pm a \sigma \sqrt{\frac 1 n + \frac{(x  \overline x)^2}{S_{X, X}}}\) is a confidence interval for \(\mu(x)\).
For \(\alpha\), we likewise have the confidence interval \(\hat a \pm a s_e \sqrt{\frac 1 n + \frac{(x  \overline x)^2}{S_{X, X}}}\).
Note that the confidence interval for \(\mu(x)\) is the confidence interval for the general population  the average for an unknown constant \(x\). However, suppose we wanted ;wip: what's the distinction between invididual vs aggregate mu?
;wip: prediction intervals
The simple linear regression model we looked at, where the response variates have Gaussian distributions that are functions of the explanatory variate, are in general called Gaussian response models.
A Gaussian response model is a model where \(Y_i \sim G(\mu(x_i), \sigma)\) for \(1 \le i \le n\). In this model, \(x_i\) is assumed to be a known constant for all \(i\), and \(E(Y_i) = \mu(x_i)\), while \(\sqrt{\mathrm{Var}(Y_i)} = \sigma\) doesn't depend on any \(x_i\).
We can also write \(Y_i = \mu(x_i) + R_i\), where \(R_i \sim G(0, \sigma)\). Here \(x_i\) is the deterministic component, while \(R_i\) is the stochastic/error component. For model checking we mostly care about the error component.
Linear regression models are Gaussian response models where \(\mu(x_i)\) has the form \(\beta_0 + \sum \beta_i x_i\). Here, the values of \(\beta_i\) are regression coefficients, and the explanatory variates \(x_i\) are also known as covariates.
Linear regression models assume that:
We must always check the model assumptions. We can check property 2 using something like a scatter plot, but what about the other two? For these we can use residuals  \(\hat r_i = y_i  \hat \mu_i\), where \(\hat \mu_i = \hat \alpha + \hat \beta_i x_i\)
The idea behind residuals is that they're observations drawn from \(R_i\), the error variables. These residuals are supposed to behave like a random variable \(R_i \sim G(0, \sigma)\). Therefore, the sum of the residuals should be 0.
Additionally, a residual plot containing \(\hat r_i\) with respect to \(x_i\). If the model assumptions hold, we should see a random scattering of points in a band around the horizontal \(\hat r_i = 0\) line, with no obvious pattern. This lets us check if assumption 2 and 3 holds.
Another type of residual plot, also used to check property 2 and 3, is to scatter plot \(\hat r_i, \hat \mu_i\) together. We should also see a similar belt around the horizontal \(\hat r_i = 0\) line, with no obvious pattern.
A third type of residual plot is a QQ plot showing the sample quantiles versus the Gaussian quantiles. This lets us check if assumption 1 holds.
If there is a pattern in the Yaxis location on the plot, it suggests that the second assumption, that \(E(Y_i) = \mu(x_i)\), might not be appropriate. For example, if the residuals plot curves upward, a quadratic regression might be more appropriate.
If there is a pattern in the spread in the Yaxis on the plot, it suggests that the third assumption, that the variance is constant, might not be appropriate. For example, if the residuals plot fans out toward the right, it seems like the variance increases as the Xaxis value increases.
(Inclass bean counting experiment)
Suppose we want to study whether hand spans of female students enrolled in STAT231 in Fall 2016 is different on average from the hand spans of male students in that same class.
First, we need a model. Let \(Y_{1, i}\) be the handspan of the \(i\)th male student, and \(Y_{2, i}\) the handspan of the \(i\) female student. Based on the given histograms and QQ plots, a Gaussian model seems to be reasonable for these  \(Y_{1, i} \sim G(\mu_1, \sigma)\) and \(Y_{2, i} \sim G(\mu_2, \sigma)\).
This is called a twosample Gaussian problem  two datasets where each data point independently has a Gaussian distribution, with two different means for each dataset, and we're assuming that the variance is the same for both. This is a special case of the Gaussian response model.
A point estimator for the difference in average hand span is \(\widetilde \mu_1  \widetilde \mu_2 = \overline Y_1  \overline Y_2\). We can also take the sample variance for one of the populations to get an estimate for \(\sigma^2\) (and check our model assumptions by comparing it to the sample variance for the other population); we'd probably pick the bigger population since that would give a more precise estimate.
A pooled estimator for \(\sigma^2\) in the twosample Gaussian problem is simply a weighted average of the estimators for each population in the problem  a weighted average of the sample variances, weighted by the number of observations in each sample.
Note that \(S_p^2 = \frac{1}{n_1 + n_2  2}\left((n_1  1)S_1^2 + (n_2  1)S_2^2\right)\), where \(S_1 = \frac 1 {n_1  2} \sum (Y_{1, i}  \overline Y_1)^2\) and \(S_2 = \frac 1 {n_2  2} \sum (Y_{2, i}  \overline Y_2)^2\). This is a pooled estimator for \(\sigma^2\) for the twosample Gaussian model.
Note that the \(\frac{\overline Y_1  \overline Y_2  (\mu_1  \mu_2)}{\sigma \sqrt{\frac 1 {n_1} + \frac 1 {n_2}}} \sim G(0, 1)\), because we have a linear combination of Gaussian random variables.
Since we don't know \(\sigma\), we can replace it with an unbiased estimator to get \(\frac{\overline Y_1  \overline Y_2  (\mu_1  \mu_2)}{S_p \sqrt{\frac 1 {n_1} + \frac 1 {n_2}}} \sim t(n_1 + n_2  2)\). We can use this as a pivotal quantity to construct a confidence interval. ;wip: why is this a pivotal quantity? it's not even a function of sigma
Since \(S_p^2 = \frac{1}{n_1 + n_2  2}\left((n_1  1)S_1^2 + (n_2  1)S_2^2\right)\), we can find a pivotal quantity for \(\sigma^2\) as \(\frac{(n_1 + n_2  2)S_p^2}{\sigma^2} \sim \chi^2(n_1 + n_2  2)\).
Using the same approach as for previous pivotal quantities, and the fact that, a \(100p\) percent confidence interval is \(\overline y_1 + \overline y_2 \pm a s_p \sqrt{\frac 1 {n_1} + \frac 1 {n_2}}\), where \(P(T < a) = \frac{1 + p}{2}\) and \(T \sim t(n_1 + n_2  2)\).
If \(n_1\) and \(n_2\) are large enough (at least 30 entries each), the studentt distribution approaches the Gaussian distribution, so we can approximately say that \(\frac{\overline Y_1  \overline Y_2  (\mu_1  \mu_2)}{S_p \sqrt{\frac 1 {n_1} + \frac 1 {n_2}}} \sim G(0, 1)\).
The bean experiment from yesterday is a paired experiment, because observations were deliberately paired up to eliminate factors like finger size, agility, and so on (each left hand observation was paired with the right hand observation from the same person). This reduces the variance (compared to, say, mixing all of the observations up) whenever the covariance is positive between the two samples. We don't do this if the covariance is negative, because then they're negatively correlated, so the resulting variance would actually be higher.
For twosample models, we often care about \(Y_1  Y_2\). Since \(Y_1  Y_2\) is a single sample, we can then apply singlesample model techniques. For our bean experiment, since \(Y_1 \sim G(\mu_1, \sigma)\) and \(Y_2 \sim G(\mu_2, \sigma)\), \(\overline Y_1  \overline Y_2 \sim G(\mu_1 + \mu_2, \sigma \sqrt{\frac 1 {n_1} + \frac 1 {n_2}})\).
The result of the experiment showed that there's strong evidence against the null hypothesis that the true value of the average bean counts are the same between both hands. This difference might be of practical significance if dealing with things like playing an instrument, designing hand tools, and so on.
Paired experiments can only be used in situations in observational experiments, because then the order in which the observations are taken would affect the result.
Suppose we want to determine if the distribution of the colour of Smarties is uniform. We buy hundreds of boxes of Smarties, and then count their colors. How do we determine how well the resulting data confirms/disconfirms the hypothesis that the colour distribution is uniform?
Clearly, the number each colour can be represented as \(Y_i \sim \mathrm{Binomial}(n, \theta_i)\). All together, they form a multinomial distribution \(Y_1, \ldots, Y_k \sim \mathrm{Multinomial}(n; \theta_1, \ldots, \theta_k)\).
So the likelihood function is \(L(\theta_1, \ldots, \theta_k; y_1, \ldots, y_k) = \theta_1^{y_1} \cdots \theta_k^{y_k}\), with all the constant factors removed. Note that \(0 \le \theta \le 1\) and \(\sum \theta_i = 1\). If we take the maximum likelihood estimate, we get \(\theta_i = \frac{y_i}{n}\).
Our null hypothesis is \(\theta_1 = \ldots = \theta_k = \frac 1 n\), or \(\theta = \theta_0\) where \(\theta_0 = \tup{\theta_1, \ldots, \theta_k}\).
To test how good this hypothesis is, we now need a test statistic. For this, we can use the likelihood ratio test statistic, \(A = 2 \log \frac{L(\theta_0)}{L(\widetilde \theta)} \sim \chi^2(k  1)\) (we have \(k\) variables \(\theta_1, \ldots, \theta_k\), but since \(\sum \theta_i = 1\), there's actually only \(k  1\) unknown values). If we expand \(\frac{L(\theta_0)}{L(\widetilde \theta)}\), we get \(\left(\frac{E_i}{Y_i}\right)^{Y_i}\), where \(E_i = \frac{;wip: get this from slides}\) is the expected frequency of successes for \(i\). So \(A = 2 \sum Y_i \ln \frac{Y_i}{E_i}\).
This test statistic makes sense because it has large values when the hypothesis isn't supported, and small values when it is. Using this test statistic, we have the pivotal quantity \(\lambda = 2 \sum y_i \ln \frac{Y_i}{E_i} \sim \chi^2(1)\). We can now generate a confidence interval or pvalue from this.
Before the likelihood ratio statistic was around, there was also the Pearson Chisquared Goodness of Fit Statistic, \(D = \sum \frac{(Y_i  E_i)^2}{E_i}\), where \(E_i = n \theta_i\)  the expected frequencies.
Alpha particle example.
For an approximate pvalue, the pvalue is around \(P(W \ge \lambda)\), where \(W \sim \chi^2(k  1  p)\), where \(p\) is the number of parameters being estimated assuming the null hypothesis is true.
For bivariate categorical data, we used relative risk to summarize this data.
Exam will include a 2way table question.
The generalized twoway table has the model \(Y_{1, 1}, Y_{1, 2}, \ldots, Y_{a, b} \sim \mathrm{Multinomial}(n; \theta_{1, 1}, \theta_{1, 2}, \ldots, \theta_{a, b})\)
In the twoway table, the null hypothesis is usually that the variables are independent.
Each \(e_{i,j} = \frac{r_i c_i}{n}\), where \(r_i\) is the row value and \(c_i\) is the column value.
For a twoway table, the number of degrees of freedom is \(ab  1  (a  1 + b  1)\).
What does it means to say that X causes Y? We usually mean that if we change X, we cause a change in Y, but how do we determine when this is the case  how do we know smoking causes lung cancer, versus lung cancer causing smoking? Does your final STAT230 grade cause your final STAT231 grade? In a lot of cases, causation is hard to define.
One definition of a causal relationship is: a variate \(x\), has a causal effect on a random variable \(Y\) if and only if changing X gets a change in Y, all other factors being held constant. However, a change in X might only change the distribution of Y, and this should still be considered a causal relationship.
A better definition is to say that changing the variate \(x\) should result in a change in some property of the random variable \(Y\), like \(E(Y)\) or \(P(Y \ge c)\).
It's usually not possible to hold all other factors constant as per the definition, but we should design studies so that all of the alternative explanations we can think of for \(Y\) changing are ruled out. Generally, we can only do this using experimental studies, so we can actually change the explanatory variate \(x\).
Actually, there are ways to detect causal relationships in observational studies. Usually we can't do this because there are multiple other reasons we could have gotten the relationship we saw:
If we can do an observational study where we control all of these confounding and lurking variates, we can establish a causal relationship. However, it's often not ethical to do so. To correct for these factors, we can instead use randomization.
If we have a large enough group, we can assume that the levels of confounding variates are about the same between the groups. We need to ensure the following hold:
My previous STAT231 notes follow:
There are 3 tutorial tests, 2 midterms, and a final exam.
The Challenger space shuttle disaster occurred due to a failed Oring. However, the right data analysis could have presented this tragedy  there was already enough data available to find this problem, but it wasn't looked at in the right way. Statistics can help prevent this sort of error.
Kansas weather reports, in a study, were found to be accurate about 85% of the time in predicting rain. However, if they just said "it's not going to rain", they would be right 90% of the time. Likewise, ESPN pundits were found to correctly predict the results of games about 48% of the time. Statistics can help us evaluate which analysts and predictors are believable.
Statistics also allows us to evaluate relationships between two things. For example, does smoking cause lung cancer, or does lung cancer cause smoking? Are they related at all?
There are two main types of data:
A transformation is a function over a variable. A linear transformation is one of the form \(y = mx + b\). A linear transformation is also known as an affine transformation.
A coding is a transformation that converts categorical data to numerical data. For example, colors can be assigned numbers, like 0 for red, 1 for orange, and so on.
We often want to extract important information about a data set to find its properties. When we do so, we extract data summaries. We can do this numerically (like finding the mean, stddev, median, etc.) or graphically (like making a scatter plot). Numerical summaries tell us about certain fundamental properties of data sets, while graphical summaries tell us the shape of the data.
Suppose we have $100 in a bank account, with 4% interest the first year, 8% the second, and 12% the third. What is the average interest rate?
Although it would seem at first glance to be 8%, the arithmetic mean doesn't represent the actual average here. The average interest rate is the single rate such that, after those 3 years, we would have the same interest gains as we did with this bank account.
Let's denote this rate as \(x\). Then according to the interest formula, \(100(1 + x)^3 = 100 \cdot 1.04 \cdot 1.08 \cdot 1.12\).
So \((1 + x)^3 = 1.257984\) and \(x = 0.0795059\), or about 0.795%.
As it turns out, this is basically the same thing as the geometric mean of the interest rates. In this case, we use the geometric mean since the interest is defined in terms of an enponential function of \(n\).
A car drives from A to B at 40 km/h, and immediately drives back at 60 km/h. What is the average speed of the car?
Although it would seem at first glance to be 50 km/h, this is not the case, since the car takes less time driving back since it's going faster.
Let's denote the distance between A and B as \(x\), in km. Then the trip from A to B took \(\frac{x}{40}\) hours, and the trip back took \(\frac{x}{60}\) hours.
Since the total time taken is \(\frac{x}{40} + \frac{x}{60}\), the average speed is \(\frac{40\frac{x}{40} + 60\frac{x}{60}}{\frac{x}{40} + \frac{x}{60}} = \frac{2}{\frac{1}{40} + \frac{1}{60}}\), or 48 km/h.
As it turns out, this is the same thing as the harmonic mean of the speeds.
When we talk about distributions, we oftne picture them as bumps or hills  a high area in the middle, and low areas on the sides. In this picture, the tails of a distribution are the low areas on the sides. The concept of tails doesn't really generalize that well, so we'll avoid it for actual formal explanations.
To find a percentile \(\beta\), we sort the observations from low to high, and then take the value at the 1indexed position \(\beta(n + 1)\) (or the sample mean of the two values closest to the 1indexed position \(\beta(n + 1)\)).
The average/central tendency isn't always the only thing we care about. Suppose you flip a coin \(n\) times before a heads appears. If you are paid \(2^n\) dollars for doing so, at what price would you pay to play this game?
Note that a head comes up on the first trial with 1/2 probability, paying out 2 dollars, on the second trial with 1/4 probability, paying out 4 dollars, and so on.
So the expected value is \(\sum_{i = 1}^\infty i \frac 1 i = \infty\)  the expected value is infinity!
However, most people wouldn't even pay 40 dollars to play  it seems like risk/variability is also an important factor in deciding whether to play. This is called St. Peter's Paradox.
Suppose we have a dataset \(x = \set{x_1, \ldots, x_n}\). Suppose we know one observation with value \(k\) is unreliable, and we want to discard it from \(x\). Can we find the new mean and variance?
Let \(y = \set{y_1, \ldots, y_{n  1}}\) be the dataset with the unreliable observation removed.
Clearly, \(\sum y_i = \sum x_i  k\), since we removed only that one element, and since \(\sum x_i = n \overline x\), \(\sum y_i = \sum x_i  k\).
So the sample mean is \(\overline y = \frac{\sum x_i  k}{n  1}\).
To find the variance, we can use the alternate form of the variance formula, so \(s_y^2 = \frac 1 {(n  1)  1} \left(\sum y_i^2  (n  1)(\overline y)^2\right)\).
Since \(\sum y_i = \sum x_i  k\), \(\sum y_i^2 = \sum x_i^2  k^2\).
Since we already know \(\overline y\), \(s_y^2 = \frac{1}{n  2} \left(\sum x_i^2  k^2\right)  \frac{\left(\sum x_i  k\right)^2}{n  1}\).
This same technique can be used to calculate the mean and variance of a dataset after adding or removing any number of items to it, without looking at the existing elements of the dataset.
When you have a dataset with sample mean \(\overline x\) and remove an item with value \(\overline x\), the mean stays the same, while the variance increases or stays the same.
The batting champion of a baseball season is the player with the highest batting average (probability of hitting the ball in each attempt, times 1000). However, although there have been 13 people with a batting average above 400 before 1941, there have been none after that year. Why does this happen?
In every sport, players have gotten better, absolutely speaking, over time  it doesn't seem like batters are actually getting worse. However, better pitchers, fielders, and managers can make a batter's job a lot harder.
We can actually test this by looking at the batting average for individual average batters over their careers. As it turns out, the average batter's batting average stays about the same over their career. However, the variance in the average player's ball hitting rate shrinks over their careers. Since the average is lower than 400, the shrinking variances mean that the tails in the batting average distribution shrink as well (causing fewer 400+ batting averages). When we become more consistent at hitting the ball at the average rate, that means there are fewer lowperformers, but also fewer highperformers.
Suppose we have a dataset \(D = \set{\tup{x_1, y_1}, \ldots, \tup{x_n, y_n}}\) with two variables. How do we measure how much these two variables are associated (how is one variable affected by the other)?
For example, suppose we have categorical variables, \(x = \set{\text{smoker}, \text{nonsmoker}}\) and \(y = \set{\text{lung cancer}, \text{no lung cancer}}\). How do we determine the relationship between these variables, given a dataset of samples?
Relative risk is a measure of association between two categorical variables. Basically, relative risk is \(\frac{P(A \mid B)}{P(A \mid \neg B)}\)  the probability of \(A\) occurring given that \(B\) occurrs over the probability of \(A\) occurring given that \(B\) does not occur. For independent variables, relative risk is 1. The farther the relative risk is from 1, the more strongly it implies that the variables are associated.
For two categorical Boolean variables, relative risk is \(\frac{\frac{\abs{x \wedge y}}{\abs{x}}}{\frac{\abs{\neg x \wedge y}}{\abs{\neg x}}}\). Expanded into a more useful form, it becomes \(\frac{\frac{\abs{x \wedge y}}{\abs{x \wedge y} + \abs{x \wedge \neg y}}}{\frac{\abs{\neg x \wedge y}}{\abs{\neg x \wedge y} + \abs{\neg x \wedge \neg y}}}\).
The sign of \(r_{xy}\) (specifically, the sign of the numerator) tells us the direction of the association (positive means variables tend to increase each other, while negative means one tends to increase when the other decreases, and vice versa), and the magnitude of \(r\) tells us the strength of the association (0 meaning no association). The denominator ensures that \(1 \le r_{x, y} \le 1\) for any dataset.
For nonlinear relationships, the sample correlation coefficient doesn't work so well. For example, for two variables \(x, y\) associated by \(y = x^2\), the sample correlation coefficient is 0, even though we defined them to be associated by a particular function. A general measure of association for any function (not just linear ones) wouldn't really be meaningful since it's always possible to construct a function to fit a finite dataset. Therefore, all measures of association must be against a particularly chosen class of functions.
For a perfect linear relationship between variables \(x, y\), like \(y = ax + b\), \(r_{xy} = 1\) if \(b > 0\), and \(r_{xy} = 1\) if \(b < 0\). If \(b = 0\), then \(r_{xy} = 0\).
Since we're assuming our dataset is a sample of the population rather than the population itself, we can only find evidence of associations rather than actual associations themselves. Even if the evidence is very strong, it doesn't say for sure that the association is there. Basically, correlation doesn't imply causation.
The reason we have summaries is to figure out the shape of a dataset, and see if we can identify it as following a certain distribution. obtaining the distribution allows us to make predictions about future observations and other useful statistical things.
The five number summary of a dataset is a common set of summaries: minimum, first quartile, median, third quartile, and, maximum. Basically, it gives 5 equally spaced points on the histogram.
Since \(\sum (x_i  \overline x)(y_i  \overline y) = \sum x_i y_i  n \overline x \overline y\) and \(\sum (x_i  \overline x)\), we can write the formula for the correlation coefficient more simply as \(\frac{\sum x_i y_i  n \overline x \overline y}{s_x s_y}\).
One of the most commonly used graphical summaries is the histogram. This is generally used when we have data that can be grouped/binned (organized into disjoint, ordered sets). By "histogram", most people actually mean a frequency histogram  a bar plot where the X axis is the bins, and the Y axis is the frequency of observations in each bin.
In contrast, statisticians usually mean density histograms when talking about histograms. This is the same thing, except the Y axis is the density of the bin.
The bins in a density histogram don't all have to all have the same range  one can encompass 30 elements, while another might encompass 10. On the plot, the bars don't all have to have the same width. The height of each bar is chosen such that the area of the bar (width of the bin times the height) is equal to the relative frequency of observations that fall into its corresponding bin. The relative frequency is simply \(\frac{\text{number of observations in the bin}}{\text{total number of observations}}\).
Note that the total area underneath the density histogram is always 1. Since the total area under a probability density function is also 1, the density histogram is very useful for comparing data with known probability density functions, which we do very often when trying to figure out if a dataset follows a certain distribution. We can't use the usual frequency histogram for this because the total area of the bars doesn't add up to 1.
The empirical cumulative distribution function (empirical CDF) for discrete numerical data is a plot where the X axis is the range of possible observation values, and the Y axis is the number of observations that are less than or equal to that value. So for a dataset \(x = \set{x_1, \ldots, x_n}\), and \(F(x_i) = \abs{\set{v \in x \middle v \le x_i}}\), the empirical CDF is a plot of \((x_i, F(x_i))\).
The advantage of the empirical CDF is that the percentile values are directly shown on the plot  to find the \(n\)th percentile, we find the X axis value such that the Y axis is \(n\) percent of the maximum Y axis value. For example, to find the median, we would find the Y axis . Also, the mode is the bar that has the largest increase in height compared to the bar on its left.
The QQ plot is used for checking if a dataset resembles a normal distribution. This is a plot of the \(\alpha\)th quantile of the dataset and the \(\alpha\)th quantile of the Z distribution \(N(0, 1)\) (normal distribution with mean 0 and variance 0), for all \(\alpha\).
So the median of the dataset is plotted at the X axis value that is the median of the Z distribution, the 95th percentile of the dataset is plotted at the X axis value that is the 95th percentile of the Z distribution, and so on. Basically, this is a plot of \((p \text{th percentile of the Z distribution}, p \text{th percentile of the dataset})\) for all \(0 \le p \le 100\). The X axis goes on infinitely in both directions.
If the QQ plot resembles a straight line, the dataset resembles a normal distribution. For example, for a normally distributed dataset, it would always be the case that the \(p\)th percentile of the dataset be a linear function of the \(p\)th percentile of the Z distribution, so all \((p \text{th percentile of the Z distribution}, p \text{th percentile of the dataset})\) would lie along the same line. For a uniform dataset, we would have a sigmoidlike shape, because the uniform distribution has no tails. For an exponential distribution, we would have a Ushaped upward curve, because the exponential distribution has a thick right tail. A distribution with a thick left tail would have a QQ plot that looks like a Ushaped downward curve.
A scatter plot plots two variables against each other, where the X axis is one variable and the Y axis is the other, and points are plotted for each observation. The scatter plot is great for finding patterns in the data, like correlations, grouping, and so on.
Quiz about numerical and graphical summaries, I'm writing in DC1351 on Thursday at 3:30PM, for just under 1 hour.
A common statistical problem is "Given a population of observations, some of the characteristics of which are unknown, and a sample taken from that population, what can we say about the population by looking just at the sample?".
The PPDAC approach is a sort of template for solving this type of problem: problem, plan, daa, analysis, and conclusion. It's specific to statistics at Waterloo.
The method of maximum likelihood is finding th most likely value of an unknown parameter of a statistical model, based on a sample. For example:
We have a coin that either has a 25% chance of getting heads, or a 75% chance of getting heads.
Suppose that we flipped the coin 100 times and got 30 heads. Obviously, we would say that it's most likely that the coin has a 25% chance of getting heads.
Mentally, it seemed like we asked "what is the probability of 30 heads occuring, if it were the case that the coin had a 25% chance of getting heads?" and "what would that probability be if it were the case that the coin had a 75% chance of getting heads instead?", and then chose the option that maximizes the probability of the event occuring.
Essentially, the method of maximum likelihood is picking the parameter that makes our observed sample properties most likely.
Formally, given a discrete distribution \(Y\) with probability function \(f\) and unknown parameter \(\theta\), then the likelihood function is defined as \(L(\theta; y_1, \ldots, y_n) = P(Y_1 = y_1, \ldots, Y_n = y_n) \text{ for the given value of } \theta\).
We then find the value of \(\theta\) that maximizes the likelihood function. This value is the maximum likelihood estimate (MLE), denoted \(\hat \theta\).
Suppose we have a variable \(X \sim \mathrm{Poisson}(\mu)\). What is the MLE of \(\mu\)?
Clearly, the probability of each sample observation is \(\frac{e^{\mu} \mu^r}{r!}\) where \(r\) is the value of the observation, by the definition of the Poisson distribution.
Therefore, \(P(Y_1 = y_1, \ldots, Y_n = y_n) = \frac{e^{\mu} \mu^{x_1}}{x_1!} \cdots \frac{e^{\mu} \mu^{x_n}}{x_n!}\) (we can just multiply the probabilities together because all of the observations are independent, which we know is the case since we're using the Poisson distribution).
We can find the value of \(\mu\) that maximizes the likelihood function by taking the logarithm of the whole thing, then finding where the derivative of that is 0. This is a pretty common way to find the max.
This value of \(\mu\) is the maximum likelihood estimate.
The midterm covers everything up to next week.
Most statistical inference problems start with parameter estimation. We generally start with a guess, and then try to refine it using samples.
A sample \(y\) can be thought of as a a set of outcomes of a random variable \(Y\), rather than just a set of numbers. This random variable is the statistical model.
To review, lowercase Greek letters like \(\mu, \theta\) are unknown parameters, lowercase Latin letters like \(x, y\) are known/sample parameters, and uppercase Latin letters like \(X, Y\) are random variables. These letters correspond: \(y_i\) is a single outcome drawn from the random variable \(Y_i\).
Likewise, \(\overline y\), the sample mean can be thought of as a single outcome of a random variable \(\overline Y\). As a result, with multiple samples, we can get a whole distribution of different \(\overline y\) values. The unknown population mean is \(\mu\), while \(\overline Y\) is the random variable with the distribution of all the different \(\overline y\) values.
These random variables, like \(\overline Y, S^2, \widetilde \pi\) are estimators. These sample values, like \(\overline y, s^2, \hat \pi\) are estimates.
Estimate the approval rating of the president:
The approval rating is \(\pi\), a population parameter that we don't know. Suppose we interview 10 people and get \(\set{A, A, D, D, D, D, A, D, D, D}\), where \(A\) means approval and \(D\) means disapproval.
We want to find \(\hat \pi\), the MLE of \(\pi\). Clearly, \(L(\pi; y_1, \ldots, y_n) = \pi \pi (1  \pi) (1  \pi) (1  \pi) (1  \pi) \pi (1  \pi) (1  \pi) (1  \pi)\), because the probability of each person approving is just \(\pi\).
So \(L(\pi; y_1, \ldots, y_n) = \pi^3 (1  \pi)^7\), and the log likelihood function is \(l(\pi) = \ln(\pi^3 (1  \pi)^7) = 3 \ln \pi + 7 \ln(1  \pi)\).
Clearly, the value of \(\pi\) that maximizes \(l(\pi)\) is \(\frac{3}{10}\) (by taking the derivative of the log likelihood function). Therefore, the MLE for the approval rating is \(\frac{3}{10}\).
Estimate the average website hits per hour, given a sample of visits over \(n\) hours \(\set{y_1, \ldots, y_n}\):
Assume that the data is from a Poisson distribution. Then we want to figure out the distribution parameter \(\mu\).
Construct the likelihood function \(L(\mu; y_1, \ldots, y_n) = \frac{e^{\mu} \mu^{y_1}}{y_1!} \cdot \ldots \cdot \frac{e^{\mu} \mu^{y_n}}{y_n!}\).
So \(L(\mu; y_1, \ldots, y_n) = \frac{e^{n\mu} \mu^{\sum y_i}}{y_1! \ldots y_n!}\).
So the log likelihood function is \(l(\mu) = n \mu + \sum (y_i \ln \mu)  \ln(y_1! \ldots y_n!)\).
Differentiating the log likelihood function with respect to \(\mu\), we get \(n + \sum \frac{y_i}{\mu}\).
By setting the derivative to 0 and solving for \(\mu\), we find the maximum value of the log likelihood function at \(\hat \mu = \frac 1 n \sum y_i = \overline y\).
So if we have a Poisson distribution, the best guess for the parameter \(\mu\) is \(\overline y\).
Estimate the probability that a Canadian contestant wins Jeopardy, given a sample of shows Canadian contestants appeared in \(\set{y_1, \ldots, y_n}\):
Clearly, if a contestant appears in \(n\) shows, then they won \(n\) in a row and lost in the \(n + 1\)th show.
We assume that shows are independent. Therefore, we have a geometric distribution, \(Y \sim \mathrm{Geo}(\pi)\).
Recall that for a geometric distribution, \(P(Y = y) = \pi^{y  1}(1  \pi)\).
So \(L(\pi; y_1, \ldots, y_n) = \pi^{y_1  1}(1  \pi) \ldots \pi^{y_n  1}(1  \pi) = \pi^{\sum y_i  n} (1  \pi)^n\).
So \(l(\pi) = (\sum y_i  n) \ln x + n \ln (1  x)\). Using the usual methods, we find the value of \(\pi\) that maximizes \(l(\pi)\) to get the MLE of \(\pi\).
Estimate the proportion of left handers in this university:
Strategy 1: Suppose we got our data by asking people from the population until we got a 10 left handers, and we needed to ask 100 before before there were 10.
Strategy 2: Suppose we got our data by asking 100 people from the population, and 10 of them turned out to be left handers.
Let \(\pi\) be the proportion of left handers. In strategy 1, we have a negative binomial distribution.
So in strategy 1, we asked 99 people and got either left or right handed, and the 100th person answered left handed. So the likelihood function is \(L(\pi) = {99 \choose 9} \pi^9 (1  \pi)^{90} \times \pi\).
So in strategy 2, we asked all 100 people, and got either left or right handed. So the likelihood function is \(L(\pi) = {100 \choose 10} \pi^{10} (1  \pi)^{90}\).
As it turns out, both strategies give the same MLE for \(\pi\), \(\frac{1}{10}\).
Note that when we're maximising value of a parameter by taking the derivative, setting it to 0, and solving for the parameter, we have to make sure that the solution is actually the global maximum, not a local maximum or even a minimum. When there are multiple solutions, we must check each of them to find the one that results in the largest likelihood value.
Note that in this course, there will only ever be one solution to the derivative being equal to 0, so we won't have to worry about it.
When we have lots of independent observations, the likelihood function will look like \(P(Y_1 = y_1) \cdot \ldots \cdot P(Y_n = y_n)\). So if \(n\) is large, we're multiplying a lot of numbers between 0 and 1, which means that the actual likelihood is going to be very small. To deal with this, we sometimes use the relative likelihood function \(R(\theta) = \frac{L(\theta)}{L(\hat \theta)}\), which always has a maximum of 1 at \(\hat \theta\).
Note that this only works for discrete distributions. For continuous ones, we have other tools.
While the likelihood interval results in a likelihood interval, the sampling distribution gives us a confidence interval.
The theory of estimation deals with problems of the form "estimate an unknown population attribute \(\theta\) given a sample drawn from the population, \(\set{y_1, \ldots, y_n}\) to get the estimate, \(\hat \theta(y_1, \ldots, y_n)\)". So far, we've looked at the maximum likelihood estimation technique for solving these for discrete distributions.
Another one is the method of least squares, where we find the value of \(\theta\) that minimises some squared error function we define  minimizing \(\sum E(y_i, \theta)^2\). If \(\theta\) is \(\mu\) in a normal distribution, then this works out to just be the sample mean, since the error is \(y_i  \mu\) and we're minizing (y_i  )^2$. This is not the best method, but it's very popular becuase it's so simple.
So far, when we're estimating we've been trying to get a single value as the estimate. However, it's often more useful to get an interval estimate instead  to find an interval \([A, B]\) such that \(\theta\) has at least a given probability of being within this interval. This type of interval problem often shows up as determining margins of error.
One way to obtain an interval estimate is via a relative likelihood function. Unlike the MLE method, where we find the most likely value, we're finding all values of a relative likelihood that exceed a certain probability threshold.
Basically, a \(p\)percent likelihood interval for an unknown parameter \(\theta\) is \(\set{\theta \middle R(\theta) \ge p}\), where \(R(\theta) = \frac{L(\theta)}{L(\hat \theta)}\) is the relative likelihood function and \(\hat \theta\) is the MLE. The lowest and highest values in the resulting set is then the likelihood interval. The likelihood interval itself, though, is a set of all values of \(\theta\) that satisfy the criteria.
All \(p\)percent likelihood intervals are subsets of \(q\)percent intervals if \(q \le p\)  smaller percentage likelihood intervals are subsets of higher percentage likelihood itnervals. The MLE belongs to all likelihood intervals. The higher the value of \(p\), the higher the probability that the true value doesn't fall into the interval. Likelihood intervals are not easy to interpret and don't have an intuitive meaning, so we generally want to use something else.
The method of sampling distributions is a better way to estimate intervals. An \(n\)percent confidence interval means that we're \(n\) percent sure that the value falls withint this interval, which is a lot more intuitive. Given a sample \(\set{y_1, \ldots, y_n}\) and a confidence threshold \(p\):
;wip: for a normal dist the coverage interval is \((\overline Y  Z^* \frac{\sigma}{\sqrt{n}}, \overline Y + Z^* \frac{\sigma}{\sqrt{n}})\), and the confidence interval is \((\overline y  Z^* \frac{\sigma}{\sqrt{n}}, \overline y + Z^* \frac{\sigma}{\sqrt{n}})\)
The confidence interval tells us the probability that the true value falls within a certain range. However, we can also use them to say "the true value must be at most \(A\) from our estimate" by choosing our sample size \(n\) such that \(A = \frac{Z^* \sigma}{\sqrt{n}}\).
The problem with confidence intervals is if we don't know \(\sigma\) or if the population isn't normal.
From the central limit theorem, if \(n\) is large, and \(Y_1, \ldots, Y_n\) are arbitrary distributions with mean \(\mu\) and variance \(\sigma^2\), then \(\overline Y \sim N(\mu, \frac{\sigma^2}{n})\).
;wip: missed first half due to interviews
Continuing with multiple examples of choosing sample sizes such that we have a certain margin of error  making the confidence interval at least a certain size. Interestingly, to get a margin of error of 3% in any population, all we need to do is ask around 1000 people, regardless of how large the population is.
I dropped this course due to my injuries, as it became infeasible to continue going to classes.