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+\definecolor{light}{rgb}{.337,.608,.741} +\definecolor{dark}{rgb}{.337,.608,.741} + +\usepackage[colorlinks=false,pdfborder={0 0 0},urlcolor= dark,colorlinks=true,linkcolor=black]{hyperref} + +\newcommand{\light}[1]{\textcolor{light}{\textbf{#1}}} +\newcommand{\dark}[1]{\textcolor{dark}{#1}} +\newcommand{\gray}[1]{\textcolor{gray}{#1}} + +%\date{} % Activate to display a given date or no date + +% + +\begin{document} + +{\LARGE \textcolor{oiB}{Learning Objectives \hfill Chapter 1: Introduction to Data}} \\ + +\begin{enumerate} +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} +\item Identify variables as numerical and categorical. +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{{\tiny $\blacksquare$}}}} +\item If the variable is numerical, further classify as continuous or discrete based on whether or not the variable can take on an infinite number of values or only non-negative whole numbers, respectively. +\item If the variable is categorical, determine if it is ordinal based on whether or not the levels have a natural ordering. +\end{itemize} + +\item Define dependent variables as variables that show some relationship with one another. Further categorize this relationship as positive or negative association, when possible. + +\item Define variables that are not associated as independent. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 1.1 and 1.2 of OpenIntro Statistics +\item Video: \href{http://youtu.be/haug_BjhgE0}{Types of variables}, Dr. \c{C}etinkaya-Rundel (3:53) +\item Test yourself: Give one example of each type of variable you have learned. \\ +\end{itemize} +}} + +% + +\vspace{0.5cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} +\item Classify a study as observational or experimental, and determine and explain why whether the study's results can be generalized to the population and whether they suggest correlation or causation. +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{{\tiny $\blacksquare$}}}} +\item If random sampling has been employed in data collection, the results should be generalizable to the target population. +\item If random assignment has been employed in study design, the results suggest causality. +\end{itemize} +\item Question confounding variables and sources of bias in a given study. +\item Distinguish between simple random, stratified, and cluster sampling, and recognize the benefits and drawbacks of choosing one sampling scheme over another. +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{{\tiny $\blacksquare$}}}} +\item Simple random sampling: Each subject in the population is equally likely to be selected. +\item Stratified sampling: First divide the population into homogenous strata (subjects within each stratum are similar, across strata are different), then randomly sample from within each strata. +\item Cluster sampling: First divide the population into clusters (subjects within each cluster are non-homogenous, but clusters are similar to each other), then randomly sample a few clusters, and then randomly sample from within each cluster. +\end{itemize} +\item Identify the four principles of experimental design and recognize their purposes: control any possible confounders, randomize into treatment and control groups, replicate by using a sufficiently large sample or repeating the experiment, and block any variables that might influence the response. +\item Identify if single or double blinding has been used in a study. +\end{enumerate} + +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Sections 1.3 - 1.5 of OpenIntro Statistics +\item Videos: +\begin{itemize} +\item[-] \href{http://youtu.be/S6y4QV7Kyl4}{Random sampling vs. random assignment}, Dr. \c{C}etinkaya-Rundel (3:18) +\item[-] \href{http://www.youtube.com/watch?v=5zyruPbgxyM}{Quick video on correlation vs. causation}, YouTube (2:19) +\item[-] \href{http://www.khanacademy.org/math/statistics/v/correlation-and-causality}{Slower video on correlation vs. causation}, KhanAcademy (10:45) +\end{itemize} +\item Article: \href{http://www.scientificamerican.com/article.cfm?id=how-anecdotal-evidence-can-undermine-scientific-results}{How Anecdotal Evidence Can Undermine Scientific Results, Scientific American, 2008} +\item Test yourself: +\begin{enumerate} +\item Describe when a study's results can be generalized to the population at large and when causation can be inferred. +\item Explain why random sampling allows for generalizability of results. +\item Explain why random assignment allows for making causal conclusions. +\item Describe a situation where cluster sampling is more efficient than simple random or stratified sampling. +\item Explain how blinding can help eliminate the placebo effect and other biases. \\ +\end{enumerate} +\end{itemize} +}} + +% + +\vspace{0.5cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} +\item Use scatterplots for describing the relationship between two numerical variables making sure to note the direction (positive or negative), form (linear or non-linear) and the strength of the relationship as well as any unusual observations that stand out. +\item When describing the distribution of a numerical variable, mention its shape, center, and spread, as well as any unusual observations. +\item Note that there are three commonly used measures of center and spread: +\begin{itemize} +\item[-] center: mean (the arithmetic average), median (the midpoint), mode (the most frequent observation). +\item[-] spread: spread: standard deviation (variability around the mean), range (max-min), interquartile range (middle 50\% of the distribution). +\end{itemize} +\item Identify the shape of a distribution as symmetric, right skewed, or left skewed, and unimodal, bimodoal, multimodal, or uniform. +\item Use histograms and box plots to visualize the shape, center, and spread of numerical distributions, and intensity maps for visualizing the spatial distribution of the data. +\item Define a robust statistic (e.g. median, IQR) as measures that are not heavily affected by skewness and extreme outliers, and determine when they are more appropriate measured of center and spread compared to other similar statistics. +\item Recognize when transformations (e.g. log) can make the distribution of data more symmetric, and hence easier to model. +\end{enumerate} +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 1.6 of OpenIntro Statistics +\item Videos: +\begin{itemize} +\item \href{http://youtu.be/FUIJruaneIc}{Mean, median, and mode}, Dr. \c{C}etinkaya-Rundel (1:16) +\item \href{http://youtu.be/ME_k8JY58f4}{Visualizing distributions of numerical variables}, Dr. \c{C}etinkaya-Rundel (6:00) +\item \href{http://www.khanacademy.org/math/statistics/v/reading-box-and-whisker-plots}{Reading box plots}, KhanAcademy (3:18) +\end{itemize} +\item Test yourself: +\begin{enumerate} +\item Describe what is meant by robust statistics and when they are used. +\item Describe when and why we might want to apply a log transformation to a variable. \\ +\end{enumerate} +\end{itemize} +}} + + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} +\item Use frequency tables and bar plots to describe the distribution of one categorical variable. +\item Use contingency tables and segmented bar plots or mosaic plots to assess the relationship between two categorical variables. +\item Use side-by-side box plots for assessing the relationship between a numerical and a categorical variable. +\end{enumerate} + +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 1.7 of OpenIntro Statistics +\item Video: \href{http://youtu.be/zLHunbpH5Hg}{Exploring relationships between categorical variables}, Dr. \c{C}etinkaya-Rundel (4:37) +\item Test yourself: +\begin{enumerate} +\item Interpret the plot in Figure 1.40 (page 39) of the textbook. +\item You collect data on 100 classmates, 70 females and 30 males. 10\% of the class are smokers, and smoking is independent of gender. Calculate how many males and females would be expected to be smokers. Sketch a mosaic plot of this scenario. \\ +\end{enumerate} +\end{itemize} +}} + +% + +\vspace{0.5cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} +\item Note that an observed difference in sample statistics suggesting dependence between variables may be due to random chance, and that we need to use hypothesis testing to determine if this difference is too large to be attributed to random chance. +\item Set up null and alternative hypotheses for testing for independence between variables, and evaluate data's support for these hypotheses using a simulation technique. +\end{enumerate} + +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 1.8 of OpenIntro Statistics +\item Test yourself: Explain why difference in sample proportions across two groups does not necessarily indicate dependence between the two variables involved? +\end{itemize} +}} + 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+\definecolor{oiR}{rgb}{.941,.318,.200} +\definecolor{oiG}{rgb}{.298,.447,.114} +\definecolor{oiY}{rgb}{.957,.863,0} + +\definecolor{light}{rgb}{.337,.608,.741} +\definecolor{dark}{rgb}{.337,.608,.741} + +\usepackage[colorlinks=false,pdfborder={0 0 0},urlcolor= dark,colorlinks=true,linkcolor=black]{hyperref} + +\newcommand{\light}[1]{\textcolor{light}{\textbf{#1}}} +\newcommand{\dark}[1]{\textcolor{dark}{#1}} +\newcommand{\gray}[1]{\textcolor{gray}{#1}} + + + +%\date{} % Activate to display a given date or no date + +% + +\begin{document} + +{\LARGE \textcolor{oiB}{Learning Objectives \hfill Chapter 3: Distributions of random variables}} \\ + +\begin{enumerate} +\renewcommand\labelenumi{\textcolor{oiB}{\textbf{LO \theenumi.}}} + +\item Calculate the standardized (Z) score of a data point given the mean and standard deviation of its distribution. + +\item Use the Z score to determine the percentile score of a data point if the distribution is normal (using technology or normal probability tables), or to assess whether or not the particular observation would be considered unusual (regardless of the shape of the distribution). + +\item Depending on the shape of the distribution determine whether the median would have a negative, positive, or 0 Z score. + +\item Assess whether or not a distribution is nearly normal using the 68-95-99.7\% rule or graphical methods such as a normal probability plot. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{oiB}{$\ast$}}} +\item Reading: Section 3.1 and 3.2 of OpenIntro Statistics +\item Video: \href{http://www.youtube.com/watch?v=ev463hHe544}{Normal Distribution - Finding Probabilities - Dr.\c{C}etinkaya-Rundel, YouTube, 6:04} +\item Video: \href{http://www.youtube.com/watch?v=6rAg4Y6NirU}{Normal Distribution - Finding Cutoff Points - Dr.\c{C}etinkaya-Rundel, YouTube, 4:25} +\item Additional resources: +\begin{itemize} +\item Video: \href{http://www.youtube.com/watch?v=McSFVzc8Swk&list=PL568547ACA9211CCA&index=24}{Normal distribution and 68-95-99.7\% rule, YouTube, 3:18} +\item Video: \href{http://www.youtube.com/watch?v=5v3Czc6ZK-Q&list=PL568547ACA9211CCA&index=23}{Z scores - Part 1, YouTube, 3:03} +\item Video: \href{http://www.youtube.com/watch?v=aa_deKPDgI4&list=PL568547ACA9211CCA&index=22}{Z scores - Part 2, YouTube, 4:01} +\end{itemize} +\item Test yourself: True/False: In a right skewed distribution the Z score of the median is positive. +\end{itemize} +}} + +% + +\vspace{0.25cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{oiB}{\textbf{LO \theenumi.}}} + +\item Determine if a random variable is binomial using the four conditions. + +\item Calculate the number of possible scenarios for obtaining $k$ successes in $n$ trials. + +\item Calculate probability of a given number of successes in a given number of trials using the binomial distribution. + +\item When number of trials is sufficiently large, use normal approximation to calculate binomial probabilities, and explain why this approach works. + +\end{enumerate} + +% + +\gray{ +{\it +\vspace{-0.75cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{oiB}{$\ast$}}} +\item Reading: Section 3.4 of OpenIntro Statistics +\item Video: \href{http://www.youtube.com/watch?v=tKmyzhvgudw}{Binomial Distribution - Finding Probabilities - Dr.\c{C}etinkaya-Rundel, YouTube, 8:46} +\item Additional resources: +\begin{itemize} +\item Video: \href{http://www.youtube.com/watch?v=oYeJBdCGwxk&list=PL568547ACA9211CCA&index=28&feature=plpp_video}{Binomial distribution, YouTube, 4:25} +\item Video: \href{http://www.youtube.com/watch?v=0er3EiM-bpg&list=PL568547ACA9211CCA&index=27&feature=plpp_video}{Mean and standard deviation of a binomial distribution, YouTube, 1:39} +\end{itemize} +\item Test yourself: +\begin{enumerate} +\item True/False: We can use the binomial distribution to determine the probability that in 10 rolls of a die the first 6 occurs on the 8th roll. +\item True / False: If a family has 3 kids, there are 8 possible combinations of gender order. +\item True/ False: When $n = 100$ and $p = 0.92$ we can use the normal approximation to the binomial to calculate the probability of 90 or more successes. +\end{enumerate} +\end{itemize} +}} + +\end{document} \ No newline at end of file diff --git a/04/04learn_obj.aux b/04/04learn_obj.aux new file mode 100644 index 0000000..4339268 --- /dev/null +++ b/04/04learn_obj.aux @@ -0,0 +1,17 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined 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value) or two-sided ($\mu$ $\ne$ the null value), and the choice depends on the research question. +\end{itemize} + +\item Define a p-value as the conditional probability of obtaining a sample statistic at least as extreme as the one observed given that the null hypothesis is true. +\[ \text{p-value} = \text{P(observed or more extreme sample statistic}~|~H_0 \text{ true)} \] + +\item Calculate a p-value as the area under the normal curve beyond the observed sample mean (either in one tail or both, depending on the alternative hypothesis). Note that in doing so you can use a Z score, where +\[ Z = \frac{sample~statistic - null~value}{SE} = \frac{\bar{x} - \mu_0}{SE} \] +\begin{itemize} +\item[-] Always sketch the normal curve when calculating the p-value, and shade the appropriate area(s) depending on whether the alternative hypothesis is one- or two-sided. +\end{itemize} + +\item Infer that if a confidence interval does not contain the null value the null hypothesis should be rejected in favor of the alternative. + +\item Compare the p-value to the significance level to make a decision between the hypotheses: +\begin{itemize} +\item[-] If the p-value $<$ the significance level, reject the null hypothesis since this means that obtaining a sample statistics at least as extreme as the observed data is extremely unlikely to happen just by chance, and conclude that the data provides evidence for the alternative hypothesis. +\item[-] If the p-value $>$ the significance level, fail to reject the null hypothesis since this means that obtaining a sample statistics at least as extreme as the observed data is quite likely to happen by chance, and conclude that the data does not provide evidence for the alternative hypothesis. +\item[-] Note that we can never ``accept" the null hypothesis since the hypothesis testing framework does not allow us to confirm it. +\end{itemize} + +\item Note that the conclusion of a hypothesis test might be erroneous regardless of the decision we make. +\begin{itemize} +\item[-] Define a Type 1 error as rejecting the null hypothesis when the null hypothesis is actually true. +\item[-] Define a Type 2 error as failing to reject the null hypothesis when the alternative hypothesis is actually true. +\end{itemize} + +\item Choose a significance level depending on the risks associated with Type 1 and Type 2 errors. +\begin{itemize} +\item[-] Use a smaller $\alpha$ is Type 1 error is relatively riskier. +\item[-] Use a larger $\alpha$ is Type 2 error is relatively riskier. +\end{itemize} + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.5cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 4.3 of OpenIntro Statistics +\item Videos: +\begin{itemize} +\item \href{http://www.youtube.com/watch?v=5N7L1cGCL-w}{Null and alternative hypotheses, YouTube} (2:42) +\end{itemize} +\item Test yourself: +{\small +\begin{enumerate} +\item List errors in the following hypotheses: $H_0: \bar{x} > 20$ and $H_A: \bar{x} \ge 25$ +\item What is wrong with the following statement? \\ +``If p-value is large we accept the null hypothesis since a large p-value implies that the observed difference between the null value and the sample statistic is quite likely to happen just by chance." +\item Suppose a researcher is interested in evaluating the following claim ``The average height of adult males in the US is 69.1 inches", and that she believes this is an underestimate. +\begin{enumerate} +\item How should she set up her hypotheses? +\item Explain to her, in plain language, how she should collect data and carry out a hypothesis test. +\item Suppose she collects a sample of 40 adult males, and finds a sample average of 70.2 inches, and a p-value of 0.0082. What should she conclude? +\item Interpret this p-value (as a conditional probability) in context of the question. +\item Suppose that the true average is in fact 69.1 inches, what type of an error has this researcher made? In order to avoid making such an error should she have used a smaller or a large significance level? +\end{enumerate} +\item Go back to Section 1.8 and describe the differences and similarities between the hypothesis testing procedure using simulation and using theory. Especially discuss how the calculation of the p-value changes while the definition stays the same. +\end{enumerate} +} +\end{itemize} +}} + +% + +\vspace{0.5cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Notice that sampling distributions of point estimates coming from samples that don't meet the required conditions for the CLT (about sample size, skew, and independence) will not be normal. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.5cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 4.4 of OpenIntro Statistics +\item Test yourself: \\ +{\small + Explain what is going on in Figure 4.20 of the book (page 186). +} +\end{itemize} +}} + +% + +\vspace{0.5cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Formulate the framework for statistical inference using hypothesis testing and nearly normal point estimates: +\begin{enumerate} +\item[(1)] Set up the hypotheses first in plain language and then using appropriate notation. +\item[(2)] Identify the appropriate sample statistic that can be used as a point estimate for the parameter of interest. +\item[(3)] Verify that the conditions for the CLT holds. +\item[(4)] Compute the SE, sketch the sampling distribution, and shade area(s) representing the p-value. +\item[(5)] Using the sketch and the normal model, calculate the p-value and determine if the null hypothesis should be rejected or not, and state your conclusion in context of the data and the research question. +\end{enumerate} + +\item If the conditions necessary for the CLT to hold are not met, note this and do not go forward with the analysis. (We will later learn about methods to use in these situations.) + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.5cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 4.5 of OpenIntro Statistics +\item Test yourself: \\ +{\small +In a random sample of 1,017 Americans 60\% said they do not trust the mass media when it comes to reporting the news fully, accurately, and fairly. The standard error associated with this estimate is 0.015 (1.5\%). What is the margin of error ay 95\% confidence level? Calculate a 95\% confidence interval and interpret it in context. You may assume that the point estimate is normally distributed (we'll learn how to check this later). +} +\end{itemize} +}} + +% + +\vspace{0.5cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Calculate the required sample size to obtain a given margin of error at a given confidence level by working backwards from the given margin of error. + +\item Distinguish statistical significance vs. practical significance. + +\item Define power as the probability of correctly rejecting the null hypothesis (complement of Type 2 error), and calculate power of a hypothesis test using a two step process: +\begin{enumerate} +\item[i.] Find the cutoff value(s) for the point estimate in order to be able to reject the null hypothesis at the given significance level. +\item[ii.] Find the probability of getting a random sample of size $n$ that yield the cutoff point estimate calculated in the previous step for a desired effect size. +\end{enumerate} + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.5cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 4.6 of OpenIntro Statistics (excluding Section 4.6.3) +\item Videos: +\begin{itemize} +\item[-] Slow but thorough: \href{http://www.youtube.com/watch?v=QW9_T8nrApU}{Calculating power, YouTube} (29:18) +\end{itemize} +\item Test yourself: \\ +{\small +If we want to decrease the margin of error, and hence have a more precise confidence interval, should we increase or decrease the sample size? +} +\end{itemize} +}} + +\end{document} \ No newline at end of file diff --git a/05/05learn_obj.aux b/05/05learn_obj.aux new file mode 100644 index 0000000..4339268 --- /dev/null +++ b/05/05learn_obj.aux @@ -0,0 +1,17 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline 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/dev/null +++ b/07/07learn_obj.tex @@ -0,0 +1,321 @@ +\documentclass[11pt]{article} +\usepackage[top=1.5cm,bottom=2cm,left=2cm,right= 2cm]{geometry} +\geometry{letterpaper} % ... or a4paper or a5paper or ... +%\geometry{landscape} % Activate for for rotated page geometry +\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{epstopdf} +\usepackage{amsmath} +\usepackage{multirow} +\usepackage{multicol} +\usepackage{changepage} +\usepackage{lscape} +\usepackage{enumitem} +\usepackage{ulem} +\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} + +\usepackage{xcolor} + +\definecolor{oiB}{rgb}{.337,.608,.741} +\definecolor{oiR}{rgb}{.941,.318,.200} +\definecolor{oiG}{rgb}{.298,.447,.114} +\definecolor{oiY}{rgb}{.957,.863,0} + +\definecolor{light}{rgb}{.337,.608,.741} +\definecolor{dark}{rgb}{.337,.608,.741} + +\usepackage[colorlinks=false,pdfborder={0 0 0},urlcolor= dark,colorlinks=true,linkcolor=black]{hyperref} + +\newcommand{\light}[1]{\textcolor{light}{\textbf{#1}}} +\newcommand{\dark}[1]{\textcolor{dark}{#1}} +\newcommand{\gray}[1]{\textcolor{gray}{#1}} + + +%\date{} % Activate to display a given date or no date + +\begin{document} + +{\LARGE \textcolor{oiB}{Learning Objectives \hfill Chapter 7: Simple linear regression}} \\ + + +% + +\begin{enumerate} +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Define the explanatory variable as the independent variable (predictor), and the response variable as the dependent variable (predicted). + +\item Plot the explanatory variable ($x$) on the x-axis and the response variable ($y$) on the y-axis, and fit a linear regression model +\[ y = \beta_0 + \beta_1 x,\] +where $\beta_0$ is the intercept, and $\beta_1$ is the slope. +\begin{itemize} +\item[-] Note that the point estimates (estimated from observed data) for $\beta_0$ and $\beta_1$ are $b_0$ and $b_1$, respectively. +\end{itemize} + +\item When describing the association between two numerical variables, evaluate +\begin{itemize} +\item[-] direction: positive ($x \uparrow, y \uparrow$), negative ($x \downarrow, y \uparrow$) +\item[-] form: linear or not +\item[-] strength: determined by the scatter around the underlying relationship +\end{itemize} + +\item Define correlation as the \emph{linear} association between two numerical variables. +\begin{itemize} +\item[-] Note that a relationship that is nonlinear is simply called an association. +\end{itemize} + +\item Note that correlation coefficient ($R$, also called Pearson's $R$) the following properties: +\begin{enumerate} +\item[-] the magnitude (absolute value) of the correlation coefficient measures the strength of the linear association between two numerical variables +\item[-] the sign of the correlation coefficient indicates the direction of association +\item[-] the correlation coefficient is always between -1 and 1, -1 indicating perfect negative linear association, +1 indicating perfect positive linear association, and 0 indicating no linear relationship +\item[-] the correlation coefficient is unitless +\item[-] since the correlation coefficient is unitless, it is not affected by changes in the center or scale of either variable (such as unit conversions) +\item[-] the correlation of X with Y is the same as of Y with X +\item[-] the correlation coefficient is sensitive to outliers +\end{enumerate} + +\item Recall that correlation does not imply causation. + +\item Define residual ($e$) as the difference between the observed ($y$) and predicted ($\hat{y}$) values of the response variable. +\[ e_i = y_i - \hat{y}_i \] + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 7.1 of OpenIntro Statistics +\item Videos: +\begin{itemize} +\item[-] Review: \href{http://www.youtube.com/watch?v=5zyruPbgxyM}{Correlation vs. causation}, YouTube (2:19) +\item[-] \href{http://www.youtube.com/watch?v=zPG4NjIkCjc}{Intro to Linear Regression}, YouTube (5:18) - very slow introduction to linear regression +\end{itemize} +\item Test yourself: +\begin{enumerate} +\item Someone hands you the scatter diagram shown below, but has forgotten to label the axes. Can you calculate the correlation coefficient? Or do you need the labels? +\begin{center} +\includegraphics[width=0.4\textwidth]{figures/no_labels} +\end{center} +\item A teaching assistant gives a quiz. There are 10 questions on the quiz and no partial credit is given. After grading the papers the TA writes down for each student the number of questions the student got right and the number wrong. What is the correlation of the number of questions right and wrong? \\ +Hint: Make up some data for number of questions right, calculate number of questions wrong, and plot them against each other. +\item Suppose you fit a linear regression model predicting score on an exam from number of hours studied. Say you've studied for 4 hours. Would you prefer to be on the line, below the line, or above the line? What would the residual for your score be (0, negative, or positive)? +\end{enumerate} +\end{itemize} +}} + +% + +\vspace{0.48cm} + +% +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Define the least squares line as the line that minimizes the sum of the squared residuals, and list conditions necessary for fitting such line: +\begin{enumerate} +\item[(1)] linearity +\item[(2)] nearly normal residuals +\item[(3)] constant variability +\end{enumerate} + +\item Define an indicator variable as a binary explanatory variable (with two levels). + +\item Calculate the estimate for the slope ($b_1$) as +\[ b_1 = R\frac{s_y}{s_x}, \] +where $R$ is the correlation coefficient, $s_y$ is the standard deviation of the response variable, and $s_x$ is the standard deviation of the explanatory variable. + +\item Interpret the slope as +\begin{itemize} +\item[-] ``For each unit increase in $x$, we would expect $y$ to increase/decrease on average by $|b_1|$ units" when $x$ is numerical. +\item[-] ``The average increase/decrease in the response variable when between the baseline level and the other level of the explanatory variable is $|b_1|$." when $x$ is categorical. +\item[-] Note that whether the response variable increases or decreases is determined by the sign of $b_1$. +\end{itemize} + +\item Note that the least squares line always passes through the average of the response and explanatory variables ($\bar{x},\bar{y}$). + +\item Use the above property to calculate the estimate for the slope ($b_0$) as +\[ b_0 = \bar{y} - b_1 \bar{x}, \] +where $b_1$ is the slope, $\bar{y}$ is the average of the response variable, and $\bar{x}$ is the average of explanatory variable. + +\item Interpret the intercept as +\begin{itemize} +\item[-] ``When $x = 0$, we would expect $y$ to equal, on average, $b_0$." when $x$ is numerical. +\item[-] ``The expected average value of the response variable for the reference level of the explanatory variable is $b_0$." when $x$ is categorical. +\end{itemize} + +\item Predict the value of the response variable for a given value of the explanatory variable, $x^\star$, by plugging in $x^\star$ in the in the linear model: +\[ \hat{y} = b_0 + b_1 x^\star \] +\begin{itemize} +\item[-] Only predict for values of $x^\star$ that are in the range of the observed data. +\item[-] Do not extrapolate beyond the range of the data, unless you are confident that the linear pattern continues. +\end{itemize} + +\item Define $R^2$ as the percentage of the variability in the response variable explained by the the explanatory variable. +\begin{itemize} +\item[-] For a good model, we would like this number to be as close to 100\% as possible. +\item[-] This value is calculated as the square of the correlation coefficient. +\end{itemize} + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 7.2 of OpenIntro Statistics +\item Videos: To be posted +\item Test yourself: +\begin{enumerate} +\item We would not want to fit a least squares line to the data shown in the scatterplot below. Which of the conditions does it appear to violate? +\begin{center} +\includegraphics[width=0.4\textwidth]{figures/nonconstant_var} +\end{center} +\item Derive the formula for $b_0$ given the fact that the linear model is $\hat{y} = b_0 + b_1 \times x$ and that the least squares line goes through ($\bar{x}, \bar{y}$). +\item One study on male college students found their average height to be 70 inches with a standard deviation of 2 inches. Their average weight was 140 pounds, with a standard deviation of 25 pounds. The correlation between their height and weight was 0.60. Assuming that the two variables are linearly associated, write the linear model for predicting weight from height. +\item Is a male who is 72 inches tall and who weighs 115 pounds on the line, below the line, or above the line? +\item Describe what is an indicator variable, and what levels 0 and 1 mean for such variables. +\item The model below predicts GPA based on an indicator variable (0: not premed, 1: premed). Interpret the intercept and slope estimates in context of the data. +\[ \widehat{gpa} = 3.57 - 0.01 \times premed \] +\item If the correlation between two variables $y$ and $x$ is 0.6, what percent of the variability in $y$ does $x$ explain? +\end{enumerate} +\end{itemize} +}} + +% + +\vspace{0.48cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Define a leverage point as a point that lies away from the center of the data in the horizontal direction. + +\item Define an influential point as a point that influences (changes) the slope of the regression line. +\begin{itemize} +\item[-] This is usually a leverage point that is away from the trajectory of the rest of the data. +\end{itemize} + +\item Do not remove outliers from an analysis without good reason. + +\item Be cautious about using a categorical explanatory variable when one of the levels has very few observations, as these may act as influential points. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 7.3 of OpenIntro Statistics +\item Videos: To be posted +\item Test yourself: Determine if each of the three unusual observations in the plot below would be considered just an outlier, a leverage point, or an influential point. +\begin{center} +\includegraphics[width=0.5\textwidth]{figures/outliers} +\end{center} +\end{itemize} +}} + +% + +\vspace{0.48cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Determine whether an explanatory variable is a significant predictor for the response variable using the $t$-test and the associated p-value in the regression output. + +\item Set the null hypothesis testing for the significance of the predictor as $H_0: \beta_1 = 0$, and recognize that the standard software output yields the p-value for the two-sided alternative hypothesis. +\begin{itemize} +\item[-] Note that $\beta_1 = 0$ means the regression line is horizontal, hence suggesting that there is no relationship between the explanatory and the response variables. +\end{itemize} + +\item Calculate the T score for the hypothesis test as +\[ T_{df} = \frac{b_1 - \text{null value}}{SE_{b_1}} \] +with $df = n - 2$. +\begin{itemize} +\item[-] Note that the T score has $n - 2$ degrees of freedom since we lose one degree of freedom for each parameter we estimate, and in this case we estimate the intercept and the slope. +\end{itemize} + +\item Note that a hypothesis test for the intercept is often irrelevant since it's usually out of the range of the data, and hence it is usually an extrapolation. + +\item Calculate a confidence interval for the slope as +\[ b_1 \pm t^\star_{df} SE_{b_1}, \] +where $df = n - 2$ and $t^\star_{df}$ is the critical score associated with the given confidence level at the desired degrees of freedom. +\begin{itemize} +\item[-] Note that the standard error of the slope estimate $SE_{b_1}$ can be found on the regression output. +\end{itemize} + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 7.4 of OpenIntro Statistics +\item Videos: To be posted +\item Test yourself: +\begin{enumerate} +\item Given the regression output below for predicting $y$ from $x$ where $n = 100$, confirm the T score and the p-value, determine whether $x$ is a significant predictor of $y$, and interpret the p-value in context. +\begin{center} +\begin{tabular}{rrrrr} + \hline + & Estimate & Std. Error & t value & Pr($>$$|$t$|$) \\ + \hline +(Intercept) & 19.8581 & 6.1379 & 3.24 & 0.0017 \\ + x & 0.2557 & 0.1055 & 2.42 & 0.0172 \\ + \hline +\end{tabular} +\end{center} +\item Calculate a 95\% confidence interval for the slope given above. +\end{enumerate} +\end{itemize} +}} + + + +\vspace{0.48cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Calculate a confidence interval for the average (expected) value of the response variable, $E(y)$, for a given $x^\star$ (within the range of the observed data) as +\[ \hat{y} \pm t^\star_{n - 2} s \sqrt{\frac{1}{n} + \frac{(x^\star - \bar{x})^2}{\sum(x_i - \bar{x})^2} } \] +where $s = \sqrt{MSE}$ + +\item Calculate a prediction interval for the predicted value of the response variable, $\hat{y}$, for a given $x^\star$ (within the range of the observed data) as +\[ \hat{y} \pm t^\star_{n - 2} s \sqrt{1 + \frac{1}{n} + \frac{(x^\star - \bar{x})^2}{\sum(x_i - \bar{x})^2} } \] +where $s = \sqrt{MSE}$ + +\item A prediction interval is similar in spirit to a confidence interval, except that the prediction interval is designed to cover a ``moving target", the random future value of $y$, while the confidence interval is designed to cover the ``fixed target", the average (expected) value of $y$, $E(y)$, for a given $x^\star$. + +\item Although both are centered at $\hat{y}$, the prediction interval is wider than the confidence interval, for a given $x^\star$ and confidence level. This makes sense, since the prediction interval must take account of the tendency of $y$ to fluctuate from its mean value, while the confidence interval simply needs to account for the uncertainty in estimating the mean value. + +\item For a given data set, the error in estimating $E(y)$ and $\hat{y}$ grows as $x^\star$ moves away from $\bar{x}$. Thus, the further $x^\star$ is from $\bar{x}$, the wider the confidence and prediction intervals will be. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Video: \href{http://youtu.be/qVCQi0KPR0s}{Prediction and confidence intervals - Dr. \c{C}etinkaya-Rundel}, YouTube (10:17) +\item Test yourself: Describe why prediction intervals are wider than confidence intervals, and why both get wider as we move away from the center of the data. +\end{itemize} +}} + + + +\end{document} \ No newline at end of file diff --git a/07/figures/.Rhistory b/07/figures/.Rhistory new file mode 100644 index 0000000..f0730f8 --- /dev/null +++ b/07/figures/.Rhistory @@ -0,0 +1,434 @@ +q() +dim(ames) +download.file("http://www.openintro.org/stat/data/ames.RData", destfile = "ames.RData") +load("ames.RData") +plt(ames) +counts <- matrix(c(1982, 10010, 11992, 4024, 990, 5014, 6006, 11000, 17006), ncol=3, nrow=3)# +colnames(counts) <- c('YF and survived','YF and died','Total')# +rownames(counts) <- c('Caucasians', 'Non-Caucasians', 'Total') +counts +counts <- matrix(c(1982, 10010, 4024, 990, 6006, 11000), ncol=2, nrow=2)# +colnames(counts) <- c('YF and survived','YF and died')# +rownames(counts) <- c('Caucasians', 'Non-Caucasians') +counts +?margin.table +margin.table(counts,1) +margin.table(counts,2) +addmargins(counts) +prop.table(counts) +prop.table(counts,1) +prop.table(counts,2) +?prop.table +prop.table(counts, margin = 1) +prop.table(counts, margin = 2) +6/34 +400/300*273 +38+14 +52+13 +88.8/7 +92.9/7 +q() +Kelsey2t <- read.csv("~/Desktop/Kelsey2t.csv") +View(Kelsey2t) +Kelsey2t = read.csv("Kelsey2t.csv") +mental <- Kelsey2t$TCU006A +summary(mental) +male <- (Kelsey2t, Kelsey2t$RANDSEX == 1) +fix(inference) +CPS4.1 <- read.csv("~/Desktop/CPS4-1.csv") +View(CPS4.1) +mean(CPS4$pop.Cath, na.rm = TRUE) +mean(CPS4.1$pop.Cath, na.rm = TRUE) +class(CPS4.1$pop.Cath) +CPS4$pop.Cath[CPS4$Nation == "Vatican"] = 99 +CPS4.1$pop.Cath[CPS4$Nation == "Vatican"] = 99 +CPS4.1$pop.Cath[CPS4.1$Nation == "Vatican"] = 99 +CPS4.1$pop.Cath[CPS4.1$Nation == "Vatican City"] = 99 +sqrt(80^2-70^2) +q() +energyuse2 <- read.csv("~/Desktop/energyuse2.csv") +View(energyuse2) +`dotPlot` <-# +function(x, fact=NULL, vertical=FALSE, at=1, key=NULL, pch=20, col=fadeColor('black', '66'), cex=1.5, add=FALSE, axes=TRUE, xlim=NULL, ylim=NULL, ...){# +skipOut <- FALSE# +if(!is.null(fact[1])){# +if(!is.null(key[1])){# +uFact <- key# +} else {# +uFact <- unique(fact)# +}# +n <- length(uFact)# +if(length(at) == 1){# +at <- 1:n# +}# +dataR <- range(x)# +atR <- range(at)# +if(!vertical){# +if(is.null(xlim[1])){# +xlim <- dataR# +}# +if(is.null(ylim[1])){# +ylim <- atR + c(-1,1)*diff(atR)/7# +}# +} else {# +if(is.null(xlim[1])){# +xlim <- atR + c(-1,1)*diff(atR)/7# +}# +if(is.null(ylim[1])){# +ylim <- dataR# +}# +}# +if(length(pch) == 1){# +pch <- rep(pch, length(at))# +}# +if(length(col) == 1){# +col <- rep(col, length(at))# +}# +if(length(cex) == 1){# +cex <- rep(cex, length(at))# +}# +dotPlot(x[fact == uFact[1]], vertical=vertical, at=at[1], axes=axes, add=add, pch=pch[1], col=col[1], cex=cex[1], xlim=xlim, ylim=ylim, ...)# +for(i in 2:n){# +dotPlot(x[fact == uFact[i]], vertical=vertical, at=at[i], add=TRUE, pch=pch[i], col=col[i], cex=cex[i], axes=FALSE)# +}# +if(axes & !add){# +if(vertical){# +axis(1, at=at, labels=uFact)# +} else {# +axis(2, at=at, labels=uFact)# +}# +}# +skipOut=TRUE# +}# +y <- rep(at[1], length(x))# +if(vertical & !skipOut){# +if(add){# +points(y,x, pch=pch[1], col=col[1], cex=cex[1], ...)# +} else {# +if(is.null(xlim[1])){# +xlim <- at[1]+c(-1,1)# +}# +if(is.null(ylim[1])){# +ylim <- range(x)# +}# +plot(y,x,axes=FALSE,xlim=xlim, ylim=ylim, pch=pch[1], col=col[1], cex=cex[1], ...)# +}# +} else if(!skipOut){# +if(add){# +points(x,y, pch=pch[1], col=col[1], cex=cex[1], ...)# +} else {# +if(is.null(ylim[1])){# +ylim <- at[1]+c(-1,1)# +}# +if(is.null(xlim[1])){# +xlim <- range(x)# +}# +plot(x,y,axes=FALSE,xlim=xlim, ylim=ylim, pch=pch[1], col=col[1], cex=cex[1], ...)# +}# +}# +if(axes & !add & is.null(fact[1])){# +axis(vertical+1)# +}# +} +q() +DataSet <- read.csv("~/Desktop/DataSet.txt") +View(DataSet) +source("http://stat.duke.edu/courses/Fall12/sta101.001/labs/inference.R") +inference(data = DataSet$pop.Cath, group = DataSet$pop.Prot, est = "mean", type = "ht", alternative = "twosided", method = "theoretical", null = 0, siglevel = .05) +inference(data = DataSet$pop.Cath, group = DataSet$pop.Prot, est = "mean", type = "ht", alternative = "twosided", method = "theoretical", null = 0) +inference(data = DataSet$pop.Cath, group = DataSet$pop.Prot, est = "mean", type = "ht", alternative = "twosided", method = "theoretical") +class(DataSet$pop.Cath) +class(DataSet$pop.Prot) +inference(data = DataSet$Paired, est = "mean", type = "ht", alternative = "twosided", method = "theoretical", null = 0) +atheismCompare <- read.csv("~/Dropbox/OI Labs/(6) lab6 - in progress/data/atheismCompare.csv") +View(atheismCompare) +usa <- atheismCompare[atheismCompare == "United States",] +View(usa) +inference(usa$response, usa$year, success = "atheist", est = "proportion", type = "ht", method = "theoretical", null = 0, alternative = "twosided") +class(usa$response,) +class(usa$response) +class(usa$year) +inference(usa$response, as.factor(usa$year), success = "atheist", est = "proportion", type = "ht", method = "theoretical", null = 0, alternative = "twosided") +source("http://stat.duke.edu/courses/Fall12/sta101.001/labs/inference.R") +680-454 +us = c(rep("a great deal", 454), rep("not a great deal", 226)) +duke = c(rep("a great deal", 10), rep("not a great deal", 10)) +response = cbind(us,duke) +gr = rep(c("us",680), rep("duke",length(duke))) +gr = c(rep("us",680), rep("duke",length(duke))) +inference(data = response, group = gr, esp = "proportion", success = "a great deal", order = c("duke","us"), +null = 0, alternative = "twosided", type = "ht", method = "theoretical") +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +null = 0, alternative = "twosided", type = "ht", method = "theoretical") +response = cbind(us,duke) +us = c(rep("a great deal", 454), rep("not a great deal", 226)) +duke = c(rep("a great deal", 10), rep("not a great deal", 10)) +response = c(us,duke) +gr = c(rep("us",680), rep("duke",length(duke))) +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +null = 0, alternative = "twosided", type = "ht", method = "theoretical") +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +type = "ci", method = "theoretical") +n_duke = 90 +suc_duke = 60 +fail_duke = n_duke - suc_duke +us = c(rep("a great deal", 454), rep("not a great deal", 226)) +duke = c(rep("a great deal", suc_duke), rep("not a great deal", fail_duke)) +response = c(us,duke) +gr = c(rep("us",680), rep("duke",length(duke))) +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +null = 0, alternative = "twosided", type = "ht", method = "theoretical") +n_duke = 90 +suc_duke = 40 +fail_duke = n_duke - suc_duke +us = c(rep("a great deal", 454), rep("not a great deal", 226)) +duke = c(rep("a great deal", suc_duke), rep("not a great deal", fail_duke)) +response = c(us,duke) +gr = c(rep("us",680), rep("duke",length(duke))) +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +null = 0, alternative = "twosided", type = "ht", method = "theoretical") +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +type = "ci", method = "theoretical") +source("http://stat.duke.edu/courses/Fall12/sta101.001/labs/inference.R") +n_duke = 88 +suc_duke = 56 +fail_duke = n_duke - suc_duke +us = c(rep("a great deal", 454), rep("not a great deal", 226)) +duke = c(rep("a great deal", suc_duke), rep("not a great deal", fail_duke)) +response = c(us,duke) +gr = c(rep("us",680), rep("duke",length(duke))) +inference(data = response, group = gr, est = "proportion", success = "a great deal", order = c("duke","us"), +type = "ci", method = "theoretical") +AFD <- read.csv("~/Desktop/AFD.csv") +View(AFD) +source("http://stat.duke.edu/courses/Fall12/sta101.001/labs/inference.R") +inference(data = AFD$AFD, group = AFD$Income, est = "mean", type = "ht", alternative = "greater", method = "theoretical") +inference(data = AFD$AFD, group = as.factor(AFD$Income), est = "mean", type = "ht", alternative = "greater", method = "theoretical") +AFD$Income = as.factor(AFD$Income) +inference(data = AFD$AFD, group = AFD$Income, est = "mean", type = "ht", alternative = "greater", method = "theoretical") +source("http://stat.duke.edu/courses/Fall12/sta101.001/labs/inference.R") +status = c(rep("Dropped Out", 132688), rep("In School", 1972233)) +state = c(rep("Florida", 71363), rep("New.York", 61325), rep("Florida", 920141), rep("New.York", 1052092)) +set.seed(724) +dropoutdata = as.data.frame(cbind(status,state)) +rows_to_sample = sample(1:nrow(dropoutdata), 1000, replace = FALSE) +samp_data = dropoutdata[rows_to_sample, ] +head(samp_data) +inference(data = samp_data$status, group = samp_data$state, est = "proportion", type = "ci", method = "theoretical", success = "In School") +80000/12 +6658.33 -2065.53 +log(NA) +log(0) +log(0.01) +q() +29+18+13 +q() +?cancor +q() +10+15+20+30+10+5 +q() +75*2*15 +75*2*14 +2100/30 +75*15 +1125/30 +60*30 +2250-1800 +450/60 +450/75 +?sumtable +??sumtable +counts <- matrix(c(1982, 10010, 4024, 990, 6006, 11000), ncol=2, nrow=2)# +> colnames(counts) <- c('YF and survived','YF and died')# +> rownames(counts) <- c('Caucasians', 'Non-Caucasians') +counts <- matrix(c(1982, 10010, 4024, 990, 6006, 11000), ncol=2, nrow=2)# +colnames(counts) <- c('YF and survived','YF and died')# +rownames(counts) <- c('Caucasians', 'Non-Caucasians') +counts +?addmargins +?subset +q() +exp(-6) +exp(-5) +plot(x = seq(-10,10), y = log(seq(-10,10))) +log(0.0144) +log(0.0063) +x = seq(0,0.10,0.01) +mean(x) +mean(log(x)) +x = seq(0.1,0.10,0.01) +mean(x) +x = seq(0.01,0.10,0.001) +mean(x) +mean(log(x)) +exp(mean(log(x))) +x = seq(0.01,0.10,0.002) +mean(x) +x = seq(0.01,0.10,0.002) +y = seq(0.03,0.12,0.002) +mean(x) +mean(y) +mean(log(x)) +mean(log(y)) +exp(mean(log(x))) +exp(mean(log(y))) +exp(-6.1) +exp(-5.4) +a = c(0,0,0,1,1,1) +b = rnorm(6) +summary(lm(b~a)) +cor(a,b)*sd(b)/sd(a) +data() +data(sleep) +sleep +?sleep +lm(sleep$extra~as.factor(sleep$group)) +summary(lm(sleep$extra~as.factor(sleep$group))) +library(xtable) +xtable( summary(lm(sleep$extra~as.factor(sleep$group)))) +data(sleep)# +sleep$group[sleep$group == 1] = "control"# +sleep$group[sleep$group == 2] = "treatment"# +sleep$group = as.factor(sleep$group)# +xtable( summary(lm(sleep$extra~sleep$group))) +sleep$group +data(sleep) +sleep +sleep$group[sleep$group == 1] = "control" +data(sleep)# +sleep$group = as.character(sleep$group)# +sleep$group[sleep$group == 1] = "control"# +sleep$group[sleep$group == 2] = "treatment"# +sleep$group = as.factor(sleep$group)# +xtable( summary(lm(sleep$extra~sleep$group))) +?sleep +?sleep +lm(sleep$extra~sleep$group) +summary(lm(sleep$extra~sleep$group)) +download.file("http://stat.duke.edu/courses/Fall12/sta101.001/data/surveyF12_fin.rda", destfile = "survey.rda")# +load("survey.rda") +names(survey) +lm(survey$gpa~survey$premed) +summary(lm(survey$gpa~survey$premed)) +xtable(summary(lm(survey$gpa~survey$premed))) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = rnorm(x)# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "") +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "") +COL +library(openintro) +COL +points(x = 40, y = 45, col = "#4C721D", pch = 4) +points(x = 40, y = 45, col = "#4C721D", pch = 8) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150))# +points(x = 40, y = 45, col = "#4C721D", pch = 8) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), xlim = c(0,60)) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 45, col = "#4C721D", pch = 8) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 45, col = "#4C721D", pch = 8)# +points(x = 40, y = 45, col = "#F05133", pch = 15)# +points(x = 40, y = 45, pch = 3) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 45, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 3) +points(x = 130, y = 10, pch = 2) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 45, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 2) +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 45, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 6) +points(x = 130, y = 10, pch = 7) +points(x = 130, y = 10, pch = 9) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 45, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 9) +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 58, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 9) +setwd("~/Desktop/Teaching/Sta 101 - F12/Learning objectives/Unit 6/figures")# +pdf("outliers.pdf", height = 3, width = 5)# +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 58, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 9)# +dev.off() +summary(lm(survey$gpa~survey$premed)) +xtable(summary(lm(survey$gpa~survey$premed))) +names(survey) +xtable(summary(lm(survey$no_drinks_drunk~survey$gender))) +x = seq(1:100) +setwd("~/Desktop/Teaching/Sta 101 - F12/Learning objectives/Unit 6/figures")# +# +set.seed(5432)# +# +pdf("outliers.pdf", height = 3, width = 5)# +par(mar=c(2,2,1,1))# +x = seq(1:100)# +y = c(rnorm(30,20,3),rnorm(30,30,3), rnorm(40, 40, 3))# +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60))# +points(x = 40, y = 58, col = "#4C721D", pch = 8)# +points(x = 140, y = 60, col = "#F05133", pch = 15)# +points(x = 130, y = 10, pch = 9)# +dev.off() +xtable(summary(lm(y ~ x))) +x = seq(1:100)# +y = c(rnorm(30,20,5),rnorm(30,30,5), rnorm(40, 40, 5))# +xtable(summary(lm(y ~ x))) +x = seq(1:100)# +y = c(rnorm(30,20,10),rnorm(30,30,10), rnorm(40, 40, 10))# +xtable(summary(lm(y ~ x))) +x = seq(1:100)# +y = c(rnorm(30,20,20),rnorm(30,30,20), rnorm(40, 40, 20))# +xtable(summary(lm(y ~ x))) +x = seq(1:100)# +y = c(rnorm(30,20,25),rnorm(30,30,25), rnorm(40, 40, 25))# +xtable(summary(lm(y ~ x))) +x = seq(1:100)# +y = c(rnorm(30,20,40),rnorm(30,30,40), rnorm(40, 40, 40))# +xtable(summary(lm(y ~ x))) +x = seq(1:100)# +y = c(rnorm(30,20,30),rnorm(30,30,30), rnorm(40, 40, 30))# +xtable(summary(lm(y ~ x))) +plot(y~x, pch = 19, col = "#569BBD", xlab = "", ylab = "", xlim = c(0,150), ylim = c(0,60)) +plot(y~x, pch = 19, col = "#569BBD") +q() diff --git a/07/figures/no_labels.R b/07/figures/no_labels.R new file mode 100644 index 0000000..6b41eb9 --- /dev/null +++ b/07/figures/no_labels.R @@ -0,0 +1,8 @@ +setwd("~/Desktop/Teaching/Sta 101 - F12/Learning objectives/Unit 6/figures") + +pdf("no_labels.pdf", height = 3, width = 5) +par(mar=c(2,2,1,1)) 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paragraphs with an empty line rather than an indent +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{epstopdf} +\usepackage{amsmath} +\usepackage{multirow} +\usepackage{multicol} +\usepackage{changepage} +\usepackage{lscape} +\usepackage{enumitem} +\usepackage{ulem} +\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} + +\usepackage{xcolor} + +\definecolor{oiB}{rgb}{.337,.608,.741} +\definecolor{oiR}{rgb}{.941,.318,.200} +\definecolor{oiG}{rgb}{.298,.447,.114} +\definecolor{oiY}{rgb}{.957,.863,0} + +\definecolor{light}{rgb}{.337,.608,.741} +\definecolor{dark}{rgb}{.337,.608,.741} + +\usepackage[colorlinks=false,pdfborder={0 0 0},urlcolor= dark,colorlinks=true,linkcolor=black]{hyperref} + +\newcommand{\light}[1]{\textcolor{light}{\textbf{#1}}} +\newcommand{\dark}[1]{\textcolor{dark}{#1}} +\newcommand{\gray}[1]{\textcolor{gray}{#1}} + + +%\date{} % Activate to display a given date or no date + +% + +\begin{document} + +{\LARGE \textcolor{oiB}{Learning Objectives \hfill Chapter 5: Inference for numerical variables}} \\ + +% + +\begin{enumerate} +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Define observations as paired if each observation in one dataset has a special correspondence or connection with exactly one observation in the other data set. + +\item Carry out inference for paired data by first subtracting the paired observations from each other, and then treating the set of differences as a new numerical variable on which to do inference (such as a confidence interval or hypothesis test for the average difference). + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 5.1 of OpenIntro Statistics +%\item Videos: To be posted +\item Test yourself: +\begin{enumerate} +\item 20 cardiac patients' blood pressure is measured before taking a medication, and after. For a given patient, are the before and after blood pressure measurements dependent (paired) or independent? +\item A random sample of 100 students were obtained and then randomly assigned into two equal sized groups. One group went on a roller coaster while the other in a simulator at an amusement park. Afterwards their blood pressure measurements were taken. Are the measurements dependent (paired) or independent? +\end{enumerate} +\end{itemize} +}} + + +% + +\vspace{0.48cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Calculate the standard error of the difference between means of two independent samples as +\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}, \] +and use this standard error in hypothesis testing and confidence intervals comparing means of independent groups. + +\item Recognize that a good interpretation of a confidence interval for the difference between two parameters includes a comparative statement (mentioning which group has the larger parameter). + +\item Recognize that a confidence interval for the difference between two parameters that doesn't include 0 is in agreement with a hypothesis test where the null hypothesis that sets the two parameters equal to each other is rejected. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 5.2 of OpenIntro Statistics +%\item Videos: To be posted +\item Test yourself: +\begin{enumerate} +\item Describe how the two sample means test is different from the paired means test, both conceptually and in terms of the calculation of the standard error. +\item A 95\% confidence interval for the difference between the number of calories consumed by mature and juvenile cats ($\mu_{mat} - \mu_{juv}$) is (80 calories, 100 calories). Interpret this interval, and determine if it suggests a significant difference between the two means. +\end{enumerate} +\end{itemize} +}} + +% + +\vspace{0.48cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Explain why means of small samples ($n < 30$) follow the t-distribution (instead of the normal, $z$, distribution). + +\item Describe how the $t$-distribution is different from the normal distribution, and what ``heavy tail" means in this context. + +\item Note that the $t$-distribution has a single parameter, degrees of freedom, and as the degrees of freedom increases this distribution approaches the normal distribution. + +\item Use a $t$-statistic, with degrees of freedom $df = n - 1$ for inference for a population mean using data from a small sample: +\[ \text{CI:}~\bar{x} \pm t^\star_{df} SE \qquad \qquad \text{HT:}~T_{df} = \frac{\bar{x} - \mu}{SE} \] +where $SE = \frac{s}{\sqrt{n}}$. + +\item Use a $t$-statistic, with degrees of freedom $df = min(n_1 - 1, n_2 - 1)$ for inference for difference between means of two population means using data from two small samples, where $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$. + +\item Make note of the pooled standard deviation but use it in very rare circumstances where the standard deviations of the populations being compared are known to be very similar. + +\item Describe how to use the $T$ table to obtain a p-value and a critical t-score ($t^\star_{df}$) for a confidence interval . + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 5.3 and 5.4 of OpenIntro Statistics +%\item Videos: To be posted +\item Test yourself: +\begin{enumerate} +\item What is the $t^\star$ for a 95\% confidence interval for a mean, where the sample size is 13. +\item What is the p-value for a hypothesis test where the alternative hypothesis is two-sided, the sample size is 20, and the test statistic, T, is calculated to be 1.75? +\end{enumerate} +\end{itemize} +}} + +\vspace{0.48cm} + +% + +\begin{enumerate}[resume] +\renewcommand\labelenumi{\textcolor{light}{\textbf{LO \theenumi.}}} + +\item Define analysis of variance (ANOVA) as a statistical inference method that is used to determine if the variability in the sample means is so large that it seems unlikely to be from chance alone by simultaneously considering many groups at once. + +\item Recognize that the null hypothesis in ANOVA sets all means equal to each other, and the alternative hypothesis suggest that at least one mean is different. +\begin{itemize} +\item[] $H_0: \mu_1 = \mu_2 = \cdots = \mu_k$ +\item[] $H_A:$ At least one mean is different +\end{itemize} + +\item List the conditions necessary for performing ANOVA +\begin{enumerate} +\item[(1)] the observations should be independent within and across groups +\item[(2)] the data within each group are nearly normal +\item[(3)] the variability across the groups is about equal +\end{enumerate} +and check if they are met using graphical diagnostics. + +\item Recognize that the test statistic for ANOVA, the F statistic, is calculated as the ratio of the mean square between groups (MSG, variability between groups) and mean square error (MSE, variability within errors), and has two degrees of freedom, one for the numerator ($df_{G} = k - 1$, where $k$ is the number of groups) and one for the denominator ($df_{E} = n - k$, where $n$ is the total sample size). +\begin{itemize} +\item[-] Note that you won't be expected to calculate MSG or MSE from the raw data, but you should have a conceptual understanding of how they're calculated and what they measure. +\end{itemize} + +\item Describe why calculation of the p-value for ANOVA is always ``one sided". + +\item Describe why conducting many $t$-tests for differences between each pair of means leads to an increased Type 1 Error rate, and we use a corrected significance level (Bonferroni corection, $\alpha^\star = \alpha / K$, where $K$ is the e number of comparisons being considered) to combat inflating this error rate. + +\item Describe why it is possible to reject the null hypothesis in ANOVA but not find significant differences between groups as a result of pairwise comparisons. + +\end{enumerate} + +\gray{ +{\it +\vspace{-0.55cm} +\begin{itemize} +\renewcommand{\labelitemi}{{\textcolor{dark}{$\ast$}}} +\item Reading: Section 5.5 of OpenIntro Statistics +%\item Videos: To be posted +\item Test yourself: +\begin{enumerate} +\item We would like to compare the average income of Americans who live in the Northeast, Midwest, South, and West. What are the appropriate hypotheses? +\item Suppose the sample in the question above has 1000 observations, what are the degrees of freedom associated with the F-statistic? +\item Suppose the null hypothesis is rejected. Describe how we would discover which regions' averages are different from each other? Make sure to discuss how many pairwise comparisons we would need to make, and what the corrected significance level would be. +\item What visualizations are useful for checking each of the conditions required for performing ANOVA? +\end{enumerate} +\end{itemize} +}} + + +\end{document} \ No newline at end of file diff --git a/06/06learn_obj.aux b/06/06learn_obj.aux new file mode 100644 index 0000000..4339268 --- /dev/null +++ b/06/06learn_obj.aux @@ -0,0 +1,17 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} 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