They also show that these regression methods deal with confounding, mediation, and interaction of causal effects in essentially the same way. It turns out that we’ve also maximized the normal likelihood. The functional form relating x and the probability of success looks like it could be an S shape. Graduate Prerequisites: The biostatistics and epidemiology MPH core course requirements and BS723 or BS852. The book is centred around traditional statistical approaches, focusing on those prevailing in research publications. Consider false positive rate, false negative rate, outliers, parsimony, relevance, and ease of measurement of predictors. \frac{p(x)}{1-p(x)} = e^{\beta_0 + \beta_1 x} In this work, we propose a novel method for integrating multiple datasets from different platforms, levels, and samples to identify common biomarkers (e.g., genes). \(e^{\beta_1}\) is the odds ratio for dying associated with a one unit increase in x. &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ Try computing the RR at 1.5 versus 2.5, then again at 1 versus 2. p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087\\ E[\mbox{grade first years}| \mbox{hours studied}] &=& \beta_{0f} + \beta_{1f} \mbox{hrs}\\ If it guesses 90% of the positives correctly, it will also guess 90% of the negatives to be positive. G &=& 525.39 - 335.23 = 190.16\\ What we see is that the vast majority of the controls were young, and they had a high rate of smoking. Agresti, A. \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) where \(\nu\) is the number of extra parameters we estimate using the unconstrained likelihood (as compared to the constrained null likelihood). \(\beta_1\) still determines the direction and slope of the line. This course provides basic knowledge of logistic regression and analysis of survival data. && \\ The third type of variable situation comes when extra variables are included in the model but the variables are neither related to the response nor are they correlated with the other explanatory variables. \end{eqnarray*}\], \[\begin{eqnarray*} \mbox{young} & \mbox{18-44 years old}\\ p(k) &=& 1-(1-\lambda)^k\\ \end{eqnarray*}\] \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ \hat{RR} &=& \frac{\frac{e^{b_0 + b_1 x}}{1+e^{b_0 + b_1 x}}}{\frac{e^{b_0 + b_1 (x+1)}}{1+e^{b_0 + b_1 (x+1)}}}\\ G &=& 3594.8 - 3585.7= 9.1\\ Maximum likelihood estimates are functions of sample data that are derived by finding the value of \(p\) that maximizes the likelihood functions. Dunn. We will study Linear Regression, Polynomial Regression, Normal equation, gradient descent and step by step python implementation. Y_i \sim \mbox{Bernoulli} \bigg( p(x_i) = \frac{e^{\beta_0 + \beta_1 x_i}}{1+ e^{\beta_0 + \beta_1 x_i}}\bigg) \end{eqnarray*}\], \[\begin{eqnarray*} In many situations, this will help us from stopping at a less than desirable model. Recall, when comparing two nested models, the differences in the deviances can be modeled by a \(\chi^2_\nu\) variable where \(\nu = \Delta p\). tau-a: Kendall’s tau-a is the number of concordant pairs minus the number of discordant pairs divided by the total number of pairs of people (including pairs who both survived or both died). && \\ Instead of trying to model the using linear regression, let’s say that we consider the relationship between the variable \(x\) and the probability of success to be given by the following generalized linear model. \end{eqnarray*}\], \[\begin{eqnarray*} \[\begin{eqnarray*} gives the \(\ln\) odds of success . &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ Abstract: In microbiome and genomic studies, the regression of compositional data has been a crucial tool for identifying microbial taxa or genes that are associated with clinical phenotypes. \hat{p}(2) &=& 0.7996326\\ \hat{p} &=& \frac{ \sum_i y_i}{n} &=& p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)}\\ Therefore, if its possible, a scatter plot matrix would be best. \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ Provides many real-data sets in various fields in the form of examples at at the end of all twelve chapters in the form of exercises. \end{eqnarray*}\] Start with the full model including every term (and possibly every interaction, etc.). \hat{p}(2.5) &=& 0.01894664\\ \mbox{& a loglikelihood of}: &&\\ The datasets below will be used throughout this course. \mathrm{logit}(\star) = \ln \bigg( \frac{\star}{1-\star} \bigg) \ \ \ \ 0 < \star < 1 An Introduction to Categorical Data Analysis. && \\ Regardless, we can see that by tuning the functional relationship of the S curve, we can get a good fit to the data. Example 5.4 Suppose that you have to take an exam that covers 100 different topics, and you do not know any of them. Symposium sessions will address challenges not only in precision medicine but also in the ongoing COVID-19 pandemic. It may even happen that the best pair of consultants are not the most knowledgeable, as there may be two that complement each other perfectly in such a way that one knows 55 topics and the other knows the remaining 45, while the most knowledgeable does not complement anybody. \mbox{test stat} &=& G\\ P(X=1 | p = 0.15) &=& 0.368\\ Use stepwise regression, which of course only yields one model unless different alpha-to-remove and alpha-to-enter values are specified. Menard, S. 1995. The pairs would be concordant if the first individual survived and the second didn’t. &=& p^{y_1}(1-p)^{1-y_1} p^{y_2}(1-p)^{1-y_2} \cdots p^{y_n}(1-p)^{1-y_n}\\ We can show that if \(H_0\) is true, Introductory course in the analysis of Gaussian and categorical data. Select the models based on the criteria we learned, as well as the number and nature of the predictors. \hat{RR} &=& \frac{\frac{e^{b_0 + b_1 x}}{1+e^{b_0 + b_1 x}}}{\frac{e^{b_0 + b_1 (x+1)}}{1+e^{b_0 + b_1 (x+1)}}}\\ G &=& 3597.3 - 3594.8 =2.5\\ We start with the empty model, and add the best predictor, assuming the p-value associated with it is smaller than, Now, we find the best of the remaining variables, and add it if the p-value is smaller than. 0 & \mbox{ don't smoke}\\ © 2020 Springer Nature Switzerland AG. p_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} In general, the method of least squares is applied to obtain the equation of the regression line. Let’s say this is Sage who knows 85 topics. We can now model binary response variables. Notice that the directionality of the low coins changes when it is included in the model that already contains the number of coins total. Unsurprisingly, there are many approaches to model building, but here is one strategy, consisting of seven steps, that is commonly used when building a regression model. &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ Consider the HERS data described in your book (page 30); variable description also given on the book website http://www.epibiostat.ucsf.edu/biostat/vgsm/data/hersdata.codebook.txt. (SBH). \mbox{specificity} &=& 120/127 = 0.945, \mbox{1 - specificity} = FPR = 0.055\\ The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. In general, the method of least squares is applied to obtain the equation of the regression line. \[\begin{eqnarray*} e^{0} &=& 1\\ While a first course in statistics is assumed, a chapter reviewing basic statistical methods is included. ], leave one out cross validation (LOOCV) [LOOCV is a special case of, build the model using the remaining n-1 points, predict class membership for the observation which was removed, repeat by removing each observation one at a time (time consuming to keep building models), like LOOCV except that the algorithm is run. A brief introduction to regression analysis of complex surveys and notes for further reading are provided. \[\begin{eqnarray*} Write out a few models by hand, does any of the significance change with respect to interaction? WHY??? z = \frac{b_1 - \beta_1}{SE(b_1)} &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ \[\begin{eqnarray*} \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) \mathrm{logit}(\star) = \ln \bigg( \frac{\star}{1-\star} \bigg) \ \ \ \ 0 < \star < 1 Some advanced topics are covered but the presentation remains intuitive. What does that even mean? x_1 &=& \begin{cases} \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 \end{eqnarray*}\], \[\begin{eqnarray*} Note 1: We can see from above that the coefficients for each variable are significantly different from zero. Suppose also that you know which topics each of your classmates is familiar with. As done previously, we can add and remove variables based on the deviance. No, you would guess \(p=0.25\)… you maximized the likelihood of seeing your data. MackLogi.sas: uses the Mack et al. In the burn data we have 308 survivors and 127 deaths = 39,116 pairs of people. To build a model (model selection). The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. Another worry when building models with multiple explanatory variables has to do with variables interacting. book series y &=& \begin{cases} -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) &=& -2 [ \ln (L(p_0)) - \ln(L(\hat{p}))]\\ advantage of integrating multiple diverse datasets over analyzing them individually. Sage. The likelihood is the probability distribution of the data given specific values of the unknown parameters. \mbox{& a loglikelihood of}: &&\\ &=& \frac{\frac{e^{b_0}e^{b_1 x}}{1+e^{b_0}e^{b_1 x}}}{\frac{e^{b_0} e^{b_1 x} e^{b_1}}{1+e^{b_0}e^{b_1 x} e^{b_1}}}\\ \ln[ - \ln (1-p(k))] &=& \beta_0 + 1 \cdot \ln(k)\\ \end{eqnarray*}\], \[\begin{eqnarray*} Advanced Methods in Biostatistics IV - Regression Modeling Advanced Methods in Biostatistics IV covers topics in modern multivariate regression from estimation theoretic, likelihood-based, and Bayesian points of view. p(0) = \frac{e^{\beta_0}}{1+e^{\beta_0}} These methods, however, are not optimized for microbiome datasets. Cancer Linear Regression. It won’t be constant for a given \(X\), so it must be calculated as a function of \(X\). The participants are postmenopausal women with a uterus and with CHD. e^{\beta_1} &=& \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ Introductory course in the analysis of Gaussian and categorical data. The results of HERS are surprising in light of previous observational studies, which found lower rates of CHD in women who take postmenopausal estrogen. \[\begin{eqnarray*} Both techniques suggest choosing a model with the smallest AIC and BIC value; both adjust for the number of parameters in the model and are more likely to select models with fewer variables than the drop-in-deviance test. Y &\sim& \mbox{Bernoulli}(p)\\ 3rd ed. We will use Bayesian and Frequentist Regression Methods Website. \hat{p}(1.5) &=& 0.9987889\\ \[\begin{eqnarray*} H_0: && \beta_1 =0\\ \end{cases} Example 4.3 Consider a simple linear regression model on number of hours studied and exam grade. \end{eqnarray*}\], \[\begin{eqnarray*} \end{eqnarray*}\], \[\begin{eqnarray*} 1985. Also problematic is that the model becomes unnecessarily complicated and harder to interpret. X_3 = \begin{cases} -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu This is done by specifying two values, \(\alpha_e\) as the \(\alpha\) level needed to enter the model, and \(\alpha_l\) as the \(\alpha\) level needed to leave the model. 1998). \[\begin{eqnarray*} \end{eqnarray*}\], \[\begin{eqnarray*} G &=& 3597.3 - 3594.8 =2.5\\ BIOST 570 Advanced Regression Methods for Independent Data (3) Covers linear models, generalized linear and non-linear regression, and models. \end{eqnarray*}\]. p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} However, the logit link (logistic regression) is only one of a variety of models that we can use. p-value &=& P(\chi^2_1 \geq 190.16) = 0 Co-organized by the Department of Biostatistics at the Harvard T.H. The course will cover extensions of these methods to correlated data using generalized estimating equations. \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 \[\begin{eqnarray*} \(\frac{L(p_0)}{L(\hat{p})}\) gives us a sense of whether the null value or the observed value produces a higher likelihood. \ln[ - \ln (1-p(k))] &=& \ln[-\ln(1-\lambda)] + \ln(k)\\ For control purposes - that is, the model will be used to control a response variable by manipulating the values of the predictor variables. One idea is to start with an empty model and adding the best available variable at each iteration, checking for needs for transformations. &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ \end{eqnarray*}\], \[\begin{eqnarray*} The senior author, Charles E. McCulloch, is head of the Division and author of Generalized Linear Mixed Models (2003), Generalized, Linear, and Mixed Models (2000), and Variance Components (1992). We continue with this process until there are no more variables that meet either requirements. If the variable seems to be useful, we keep it and move on to looking for a second. The training set, with at least 15-20 error degrees of freedom, is used to estimate the model. &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ If you set \(\alpha_e\) to be very small, you might walk away with no variables in your model, or at least not many. \mathrm{logit}(p(x+1)) &=& \beta_0 + \beta_1 (x+1)\\ Biostatistical Methods Overview, Programs and Datasets (First Edition) ... fits the Poisson regression models using the SAS program shown in Table 8.2 that generates the output shown in Tables 8.3, 8.4 and 8.5. But more importantly, age is a variable that reverses the effect of smoking on cancer - Simpson’s Paradox. \end{eqnarray*}\] # predicting the probability of success (on the `scale` of the response variable): # do NOT use the SE to create a CI for the predicted value, # instead, use the SE from `type="link" ` and transform the interval, http://www.biostat.ucsf.edu/vgsm/data/excel/hersdata.xls, http://www.epibiostat.ucsf.edu/biostat/vgsm/data/hersdata.codebook.txt, https://onlinecourses.science.psu.edu/stat501/node/332, Categorical variable indicating level of snoring, (never=1, occasionally=2, often=3 and always=4), The response isn’t linear (until we transform), The predicted values go outside the bounds of (0,1), probability of success is constant for a particular. To maximize the likelihood, we use the natural log of the likelihood (because we know we’ll get the same answer): \end{eqnarray*}\], \[\begin{eqnarray*} P(X=1 | p = 0.5) &=& 0.25\\ \mbox{interaction model} &&\\ \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ Statistics for Biology and Health However, within each group, the cases were more likely to smoke than the controls. \mathrm{logit}(p(x)) &=& \beta_0 + \beta_1 x\\ The worst thing that happens is that the error degrees of freedom is lowered which makes confidence intervals wider and p-values bigger (lower power). A good chunk of the cases were older, and the rate of smoking was substantially lower in the oldest group. p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 How do we model? G &=& 525.39 - 335.23 = 190.16\\ When using our method, we set μ=1 and α=0.5 except LASSO penalty. If none of the models provide a satisfactory fit, try something else, such as collecting more data, identifying different predictors, or formulating a different type of model. \end{eqnarray*}\]. \[\begin{eqnarray*} This method of estimating the parameters of a regression line is known as the method of least squares. 0 & \text{otherwise} \\ which gives a likelihood of: Not logged in RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ A better strategy is to select the second not by considering what he or she knows regarding the entire agenda, but by looking for the person who knows more about the topics than the first does not know (the variable that best explains the residual of the equation with the variables entered). The big model (with all of the interaction terms) has a deviance of 3585.7; the additive model has a deviance of 3594.8. H_1: && \beta_1 \ne 0\\ i Fitting Regression Lines—The Method of Least Squares 2( )( ) 0 In particular, methods are illustrated using a variety of data sets. \[\begin{eqnarray*} &=& \sum_i y_i \ln(p) + (n- \sum_i y_i) \ln (1-p)\\ Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 There are various ways of creating test or validation sets of data: Length of Bird Nest This example is from problem E1 in your text and includes 99 species of N. American passerine birds. This is designed to be a first course in Statistics. &=& -2 [ \ln(0.0054) - \ln(0.0697) ] = 5.11\\ OR &=& \mbox{odds dying if } (x_1, x_2) / \mbox{odds dying if } (x_1^*, x_2^*) = \frac{e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2}}{e^{\beta_0 + \beta_1 x_1^* + \beta_2 x_2^*}}\\ This new book provides a unified, in-depth, readable introduction to the multipredictor regression methods most widely used in biostatistics: linear models for continuous outcomes, logistic models for binary outcomes, the Cox model for right-censored survival times, repeated-measures models for longitudinal and hierarchical outcomes, and generalized linear models for counts and other outcomes. Since each observed response is independent and follows the Bernoulli distribution, the probability of a particular outcome can be found as: Taken from https://onlinecourses.science.psu.edu/stat501/node/332. &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ A study was undertaken to investigate whether snoring is related to a heart disease. We can now model binary response variables. In general, there are five reasons one might want to build a regression model. The methods introduced include robust estimation, testing, model selection, model check and diagnostics. \mbox{middle} & \mbox{45-64 years old}\\ By using Kaggle, you agree to our use of cookies. \mbox{simple model} &&\\ p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ \end{eqnarray*}\], Let’s say the log odds of survival for given observed (log) burn areas \(x\) and \(x+1\) are: With correlated variables it is still possible to get unbiased prediction estimates, but the coefficients themselves are so variable that they cannot be interpreted (nor can inference be easily performed). p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ Supplemented with numerous graphs, charts, and tables as well as a Web site for larger data sets and exercises, Biostatistical Methods: The Assessment of Relative Risks is an excellent guide for graduate-level students in biostatistics and an invaluable reference for … RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ \mathrm{logit}(\hat{p}) &=& 22.708 - 10.662 \cdot \ln(\mbox{ area }+1)\\ That is, the difference in log likelihoods will be the opposite difference in deviances: Bayesian and Frequentist Regression Methods Website. p_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} We’d like to know how well the model classifies observations, but if we test on the samples at hand, the error rate will be much lower than the model’s inherent accuracy rate. Some are available in Excel and ASCII ( .csv) formats and Stata (.dta).Methods for retrieving and importing datasets may be found here.If you need one of the datasets we maintain converted to a non-S format please e-mail mailto:charles.dupont@vanderbilt.edu to make a request. \end{eqnarray*}\], \[\begin{eqnarray*} The second type is MetaLasso, and our proposed method is as the third type. If there are too many, we might just look at the correlation matrix. \end{eqnarray*}\], \[\begin{eqnarray*} \(\chi^2\): The Likelihood ratio test also tests whether the response is explained by the explanatory variable. 2nd ed. The explanatory variable of interest was the length of the bird. x_1 &=& \begin{cases} The logistic regression model is overspecified. For theoretical reasons - that is, the researcher wants to estimate a model based on a known theoretical relationship between the response and predictors. The rules, however, state that you can bring two classmates as consultants. E[\mbox{grade}| \mbox{hours studied}] &=& \beta_{0} + \beta_{1} \mbox{hrs} + \beta_2 I(\mbox{year=senior}) + \beta_{3} \mbox{hrs} I(\mbox{year = senior})\\ Simpson’s paradox is when the association between two variables is opposite the partial association between the same two variables after controlling for one or more other variables. Recall: \[\begin{eqnarray*} 1. Over 10 million scientific documents at your fingertips. E[\mbox{grade seniors}| \mbox{hours studied}] &=& \beta_{0s} + \beta_{1s} \mbox{hrs}\\ \end{eqnarray*}\], \[\begin{eqnarray*} When a model is overspecified, there are one or more redundant variables. Applied Logistic Regression Analysis. &=& p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)}\\ &=& -2 \Bigg[ \ln \bigg( (0.25)^{49} (0.75)^{98} \bigg) - \ln \Bigg( \bigg( \frac{1}{3} \bigg)^{49} \bigg( \frac{2}{3} \bigg)^{98} \Bigg) \Bigg]\\ Bayesian and Frequentist Regression Methods (Springer Series in Statistics) - Kindle edition by Wakefield, Jon. \mbox{overall OR} &=& e^{-0.37858 } = 0.6848332\\ \end{eqnarray*}\], \[\begin{eqnarray*} 5, 6 Undetected batch effects can have major impact on subsequent conclusions in both unsupervised and supervised analysis. \end{eqnarray*}\] However, looking at all possible interactions (if only 2-way interactions, we could also consider 3-way interactions etc. Length as a continuous explanatory variable: Length as a categorical explanatory variables: Length plus a few other explanatory variables: https://interactions.jacob-long.com/index.html. L(\underline{p}) &=& \prod_i \Bigg( \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)^{y_i} \Bigg(1-\frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)^{(1- y_i)} \\ \end{eqnarray*}\], \[\begin{eqnarray*} \mathrm{logit}(p) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 In this case, one could say that you were overfitting the past exam papers and that the knowledge gained didn’t generalize to future exam questions. It is quite common to have binary outcomes (response variable) in the medical literature. Part of Springer Nature. H_1:&& p \ne 0.25\\ \end{eqnarray*}\], \[\begin{eqnarray*} How is it interpreted? \end{eqnarray*}\], \[\begin{eqnarray*} \[\begin{eqnarray*} The validation set is used for cross-validation of the fitted model. 2 Several methods that remove or adjust batch variation have been developed. &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ X_2 = \begin{cases} Now, if the upcoming exam completely consists of past questions, you are certain to do very well. and reduced (null) models. \end{eqnarray*}\] \end{eqnarray*}\], (Suppose we are interested in comparing the odds of surviving third-degree burns for patients with burns corresponding to log(area +1)= 1.90, and patients with burns corresponding \end{eqnarray*}\], "~/Dropbox/teaching/math150/PracStatCD/Data Sets/Chapter 07/CSV Files/C7 Birdnest.csv", \[\begin{eqnarray*} \mbox{additive model} &&\\ &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ \end{eqnarray*}\]. \end{eqnarray*}\] Though it is important to realize that we cannot find estimates in closed form. p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 where we are modeling the probability of 20-year mortality using smoking status and age group. \hat{p} &=& \frac{ \sum_i y_i}{n} \end{eqnarray*}\] That is because age and smoking status are so highly associated (think of the coin example). \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ always. Using the burn data, convince yourself that the RR isn’t constant. That is, for one level of a variable, the relationship of the main predictor on the response is different. However, (Menard 1995) warns that for large coefficients, standard error is inflated, lowering the Wald statistic (chi-square) value. The results of the first large randomized clinical trial to examine the effect of hormone replacement therapy (HRT) on women with heart disease appeared in JAMA in 1998 (Hulley et al. \mbox{sensitivity} &=& TPR = 144/308 = 0.467\\ This new edition provides a unified, in-depth, readable introduction to the multipredictor regression methods most widely used in biostatistics: linear models for continuous outcomes, logistic models for binary outcomes, the Cox model for right-censored survival times, repeated-measures models for longitudinal and hierarchical outcomes, and generalized linear models for counts and other outcomes. You begin by trying to answer the questions from previous papers and comparing your answers with the model answers provided. Let’s say \(X \sim Bin(p, n=4).\) We have 4 trials and \(X=1\). (The fourth step is very good modeling practice. For now, we will try to predict whether the individuals had a medical condition, medcond (defined as a pre-existing and self-reported medical condition). \end{eqnarray*}\] Generally, extraneous variables are not so problematic because they produce models with unbiased coefficient estimators, unbiased predictions, and unbiased variance estimates. to log(area +1)= 2.00. For data summary reasons - that is, the model will be used merely as a way to summarize a large set of data by a single equation. \[\begin{eqnarray*} \mbox{test stat} &=& \chi^2\\ P(X=1 | p = 0.25) &=& 0.422\\ \[\begin{eqnarray*} -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu A: Let’s say we use prob=0.25 as a cutoff: \[\begin{eqnarray*} \end{eqnarray*}\]. \end{eqnarray*}\] The least-squares line, or estimated regression line, is the line y = a + bx that minimizes the sum of the squared distances of the sample points from the line given by . This method of estimating the parameters of a regression line is known as the method of least squares. 1 & \mbox{ smoke}\\ (Technometrics, February 2002) "...a focused introduction to the logistic regression model and its use in methods for modeling the relationship between a categorical outcome variable and a … That is, the variables are important in predicting odds of survival. Consider looking at all the pairs of successes and failures. Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt Example 5.2 The Heart and Estrogen/progestin Replacement Study (HERS) is a randomized, double-blind, placebo-controlled trial designed to test the efficacy and safety of estrogen plus progestin therapy for prevention of recurrent coronary heart disease (CHD) events in women. The logistic regression model is a generalized linear model. Collect the data. \[\begin{eqnarray*}GLM: g(E[Y | X]) = \beta_0 + \beta_1 X\end{eqnarray*}\]where \(g(\cdot)\)is the … &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ Faculty in UW Biostatistics are developing new statistical learning methods for the analysis of large-scale data sets, often by exploiting the data’s inherent structure, such as sparsity and smoothness. \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 p-value &=& P(\chi^2_1 \geq 190.16) = 0 Statistical tools for analyzing experiments involving genomic data. \ln L(\underline{p}) &=& \sum_i y_i \ln\Bigg( \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg) + (1- y_i) \ln \Bigg(1-\frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)\\ \end{cases}\\ \beta_{0s} &=& \beta_0 + \beta_2\\ To account for the variation in sequencing depth and high dimensionality of read counts, a high-dimensional log-contrast model is often used where log compositions of read counts are used as covariates. Covers all of the nuts and bolts of biostatistics in a user-friendly style that motivates readers. \end{eqnarray*}\], \[\begin{eqnarray*} How do you choose the \(\alpha\) values? This dataset includes data taken from cancer.gov about deaths due to cancer in the United States. H_1: && \beta_1 \ne 0\\ I can’t possibly over-emphasize the data exploration step. The Heart and Estrogen/Progestin Replacement Study (HERS) found that the use of estrogen plus progestin in postmenopausal women with heart disease did not prevent further heart attacks or death from coronary heart disease (CHD). 0 & \text{otherwise} \\ The majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ Note 2: We can see that smoking becomes less significant as we add age into the model. e^{\beta_1} &=& \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ \end{eqnarray*}\]. P( \chi^2_1 \geq 5.11) &=& 0.0238 That is, the variables contain the same information as other variables (i.e., are correlated!). \mbox{sensitivity} &=& TPR = 265/308 = 0.860\\ 1. \end{eqnarray*}\], \[\begin{eqnarray*} \[\begin{eqnarray*} They are: Decide which explanatory variables and response variable on which to collect the data. More on this as we move through this model. \end{cases} For example: consider a pair of individuals with burn areas of 1.75 and 2.35. Helpfully, Professor Hardin has made previous exam papers and their worked answers available online. \[\begin{eqnarray*} p(-\beta_0 / \beta_1) &=& p(x) = 0.5 P(X=1 | p = 0.05) &=& 0.171\\ The authors are on the faculty in the Division of Biostatistics, Department of Epidemiology and Biostatistics, University of California, San Francisco, and are authors or co-authors of more than 200 methodological as well as applied papers in the biological and biomedical sciences. Example 5.3 Consider the example on smoking and 20-year mortality (case) from section 3.4 of Regression Methods in Biostatistics, pg 52-53. \[\begin{eqnarray*} &=& p^{y_1}(1-p)^{1-y_1} p^{y_2}(1-p)^{1-y_2} \cdots p^{y_n}(1-p)^{1-y_n}\\ “Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions.” Journal of the American Statistical Association, 141–50. 198.71.239.51, applied regression methods for biomedical research, linear, logistic, generalized linear, survival (Cox), GEE, a, Department of Epidemiology and Biostatistics, Springer Science+Business Media, Inc. 2005, Repeated Measures and Longitudinal Data Analysis. The least-squares line, or estimated regression line, is the line y = a + bx that minimizes the sum of the squared distances of the sample points from the line given by . Study bivariate relationships to reveal other outliers, to suggest possible transformations, and to identify possible multicollinearities. \mathrm{logit}(p) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 The difference between these two probabilities, 0.00499 was discounted as being too small to worry about. 1 & \text{for always} \\ Unfortunately, you get carried away and spend all your time on memorizing the model answers to all past questions. In the table below are recorded, for each midpoint of the groupings log(area +1), the number of patients in the corresponding group who survived, and the number who died from the burns. x_2 &=& \begin{cases} \mathrm{logit}(\hat{p}) &=& 22.708 - 10.662 \cdot \ln(\mbox{ area }+1)\\ Linear Regression Datasets for Machine Learning. “Randomized Trial of Estrogen Plus Progestin for Secondary Prevention of Coronary Heart Disease in Postmenopausal Women.” Journal of the American Medical Association 280: 605–13. \end{eqnarray*}\], When each person is at risk for a different covariate (i.e., explanatory variable), they each end up with a different probability of success. \end{eqnarray*}\] \mathrm{logit}(\hat{p}) = 22.708 - 10.662 \cdot \ln(\mbox{ area }+1). \end{eqnarray*}\], \[\begin{eqnarray*} where \(\nu\) represents the difference in the number of parameters needed to estimate in the full model versus the null model. \end{cases} For predictive reasons - that is, the model will be used to predict the response variable from a chosen set of predictors. That is, a linear model as a function of the expected value of the response variable. \[\begin{eqnarray*} \end{eqnarray*}\] \end{eqnarray*}\], \[\begin{eqnarray*} Evaluate the selected models for violation of the model conditions. (Agresti 1996) states that the likelihood-ratio test is more reliable for small sample sizes than the Wald test. P(X=1 | p = 0.9) &=& 0.0036 \\ Does the interpretation change with interaction? Not affiliated Using the additive model above: Datasets Most of the datasets on this page are in the S dumpdata and R compressed save() file formats. &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 We minimized the residual sum of squares: For logistic regression, we use the logit link function: \end{eqnarray*}\] P(X=1 | p = 0.75) &=& 0.047 \\ p(x) &=& 1 - \exp [ -\exp(\beta_0 + \beta_1 x) ] The pairs would be discordant if the first individual died and the second survived. Because we will use maximum likelihood parameter estimates, we can also use large sample theory to find the SEs and consider the estimates to have normal distributions (for large sample sizes). But really, usually likelihood ratio tests are more interesting. If you could bring only one consultant, it is easy to figure out who you would bring: it would be the one who knows the most topics (the variable most associated with the answer). 0 &=& (1-p) \sum_i y_i + p (n-\sum_i y_i) \\ \ln (p(x)) = \beta_0 + \beta_1 x 0 &=& (1-p) \sum_i y_i + p (n-\sum_i y_i) \\ \mbox{old} & \mbox{65+ years old}\\ \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ 2. G &=& 2 \cdot \ln(L(MLE)) - 2 \cdot \ln(L(null))\\ (see log-linear model below, 5.1.2.1 ). There might be a few equally satisfactory models. \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 \[\begin{eqnarray*} The method was based on multitask regression model enforced with sparse group Ours is called the logit. 2012. We can use the drop-in-deviance test to test the effect of any or all of the parameters (of which there are now four) in the model. Before we do that, we can define two criteria used for suggesting an optimal model. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ The link is the relationship between the response variable and the linear function in x. We use cookies on Kaggle to deliver our services, analyze web traffic, and improve your experience on the site. gives the odds of success. Download it once and read it on your Kindle device, PC, phones or tablets. \end{eqnarray*}\], \[\begin{eqnarray*} G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} 1 & \text{for occasionally} \\ \end{eqnarray*}\], D: all models will go through (0,0) \(\rightarrow\) predict everything negative, prob=1 as your cutoff, E: all models will go through (1,1) \(\rightarrow\) predict everything positive, prob=0 as your cutoff, F: you have a model that gives perfect sensitivity (no FN!) Don’t worry about building the model (classification trees are not a topic for class), but check out the end where they talk about predicting on test and training data. Contains notes on computations at the end of most chapters, covering the use of Excel, SAS, and others. The examples, analyzed using Stata, are drawn from the biomedical context but generalize to other areas of application. \mbox{specificity} &=& 61 / 127 = 0.480, \mbox{1 - specificity} = FPR = 0.520\\ That is, the odds of survival for a patient with log(area+1)= 1.90 is 2.9 times higher than the odds of survival for a patient with log(area+1)= 2.0.). Applied Logistic Regression is an ideal choice." \mbox{middle} & \mbox{45-64 years old}\\ Note 3: We can estimate any of the OR (of dying for smoke vs not smoke) from the given coefficients: Age seems to be less important than drinking status. Includes interpretation of parameters, including collapsibility and non-collapsibility, estimating equations; likelihood; sandwich estimations; the bootstrap; Bayesian inference: prior specification, hypothesis testing, and computation; comparison of … We can estimate the SE (Wald estimates via Fisher Information). We can, however, measure whether or not the estimated model is consistent with the data. \end{eqnarray*}\], \[\begin{eqnarray*} \end{cases} \mathrm{logit}(p(x_1, x_2) ) &=& \beta_0 + \beta_1 x_1 + \beta_2 x_2\\ But if the new exam asks different questions about the same material, you would be ill-prepared and get a much lower mark than with a more traditional preparation. \end{eqnarray*}\]. (Think about Simpson’s Paradox and the need for interaction.). \end{eqnarray*}\], \[\begin{eqnarray*} p-value &=& P(\chi^2_6 \geq 9.1)= 1 - pchisq(9.1, 6) = 0.1680318 Supplemented with numerous graphs, charts, and tables as well as a Web site for larger data sets and exercises, Biostatistical Methods: The Assessment of Relative Risks is an excellent guide for graduate-level students in biostatistics and an invaluable reference for biostatisticians, applied statisticians, and epidemiologists. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. The above inequality holds because \(\hat{\underline{p}}\) maximizes the likelihood. Ramsey, F., and D. Schafer. For many students and researchers learning to use these methods, this one book may be all they need to conduct and interpret multipredictor regression analyses. \mbox{additive model} &&\\ The logistic regression model contains extraneous variables. Decide on the type of model that is needed in order to achieve the goals of the study. \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ \end{eqnarray*}\], \[\begin{eqnarray*} \ln L(p) &=& \ln \Bigg(p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)} \Bigg)\\ x_2 &=& \begin{cases} \end{eqnarray*}\], \[\begin{eqnarray*} (The logistic model is just one model, there isn’t anything magical about it. The patients were grouped according to the area of third-degree burns on the body (measured in square cm). &=& \mbox{null (restricted) deviance - residual (full model) deviance}\\ These data refer to 435 adults who were treated for third-degree burns by the University of Southern California General Hospital Burn Center. In fact, usually, we use them to test whether the coefficients are zero: \[\begin{eqnarray*} In particular, methods are illustrated using a variety of data sets. \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ \frac{p(x)}{1-p(x)} = e^{\beta_0 + \beta_1 x} GLM: g(E[Y | X]) = \beta_0 + \beta_1 X Lesson of the story: be very very very careful interpreting coefficients when you have multiple explanatory variables. The logistic regression model is correct! \end{cases} Being underspecified is the worst case scenario because the model ends up being biased and predictions are wrong for virtually every observation. L(\hat{\underline{p}}) > L(p_0) \[\begin{eqnarray*} The first type of method applied logistic regression model with the four penalties to the merged data directly. This data set was obtained (downloaded) from the SAS online data sets for the SAS book Logistic Regression Examples Using the SAS System, where it appears on p. 17-18. \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x It gives you a sense of whether or not you’ve overfit the model in the building process.) L(\underline{p}) &=& \prod_i \Bigg( \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)^{y_i} \Bigg(1-\frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)^{(1- y_i)} \\ G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} We see above that the logistic model imposes a constant OR for any value of \(X\) (and not a constant RR). [Where \(\hat{\underline{p}}\) is the maximum likelihood estimate for the probability of success (here it will be a vector of probabilities, each based on the same MLE estimates of the linear parameters). ] Once \(y_1, y_2, \ldots, y_n\) have been observed, they are fixed values. We should also look at interactions which we might suspect. However, the scatterplot of the proportions of patients surviving a third-degree burn against the explanatory variable shows a distinct curved relationship between the two variables, rather than a linear one. The second half introduces bivariate and multivariate methods, emphasizing contingency table analysis, regression, and analysis of variance. \beta_0 + \beta_1 x &=& 0\\ \hat{p} &=& \frac{49}{147}\\ \[\begin{eqnarray*} Note that the x-axis is some continuous variable x while the y-axis is the probability of success at that value of x. This method follows in the same way as Forward Regression, but as each new variable enters the model, we check to see if any of the variables already in the model can now be removed. To assess a model’s accuracy (model assessment). How do we decide? John Wiley; Sons, New York. Note 4 Every type of generalized linear model has a link function. p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 \end{eqnarray*}\] The previous model specifies that the OR is constant for any value of \(X\) which is not true about RR. \mbox{young} & \mbox{18-44 years old}\\ Tied pairs occur when the observed survivor has the same estimated probability as the observed death. &=& \frac{\frac{e^{b_0}e^{b_1 x}}{1+e^{b_0}e^{b_1 x}}}{\frac{e^{b_0} e^{b_1 x} e^{b_1}}{1+e^{b_0}e^{b_1 x} e^{b_1}}}\\ &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ For inferential reasons - that is, the model will be used to explore the strength of the relationships between the response and the predictors. \end{eqnarray*}\], \[\begin{eqnarray*} GLM: g(E[Y | X]) = \beta_0 + \beta_1 X The logistic regression model is underspecified. Consider a toy example describing, for example, flipping coins. p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} The additive model has a deviance of 3594.8; the model without weight is 3597.3. Model building is definitely an ``art." 1998. P(X=1 | p = 0.9) &=& 0.0036 \\ &=& \ln \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ The problem with this strategy is that it may be that the 75 subjects Bruno knows are already included in the 85 that Sage knows, and therefore, Bruno does not provide any knowledge beyond that of Sage. P(X=1 | p = 0.05) &=& 0.171\\ \end{eqnarray*}\], Using the logistic regression model makes the likelihood substantially more complicated because the probability of success changes for each individual. P(Y_1=y_1, Y_2=y_2, \ldots, Y_n=y_n) &=& P(Y_1=y_1) P(Y_2 = y_2) \cdots P(Y_n = y_n)\\ 1 & \text{for often} \\ Before reading the notes here, look through the following visualization. \end{eqnarray*}\]. P(X=1 | p = 0.15) &=& 0.368\\ Recall: \end{eqnarray*}\]. Biostatistics with R provides a straightforward introduction on how to analyse data from the wide field of biological research, including nature protection and global change monitoring. If you set it to be large, you will wander around for a while, which is a good thing, because you will explore more models, but you may end up with variables in your model that aren’t necessary. 1 - p(x) = \frac{1}{1+e^{\beta_0 + \beta_1 x}} \mbox{test stat} &=& \chi^2\\ Use features like bookmarks, note taking and highlighting while reading Bayesian and Frequentist Regression Methods (Springer Series in Statistics). ), things can get out of hand quickly. \[\begin{eqnarray*} \[\begin{eqnarray*} \[\begin{eqnarray*} i Fitting Regression Lines—The Method of Least Squares 2( )( ) 0 1996. The majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. \beta_{0f} &=& \beta_{0}\\ Maximizing the likelihood? A first idea might be to model the relationship between the probability of success (that the patient survives) and the explanatory variable log(area +1) as a simple linear regression model. \end{eqnarray*}\], \[\begin{eqnarray*} The output generated differs slightly from that shown in the tables. L(\underline{y} | b_0, b_1, \underline{x}) &=& \prod_i \frac{1}{\sqrt{2 \pi \sigma^2}} e^{(y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ The effect is not due to the observational nature of the study, and so it is important to adjust for possible influential variables regardless of the study at hand. [\(\beta_1\) is the change in log-odds associated with a one unit increase in x. \end{eqnarray*}\]. \end{eqnarray*}\], \[\begin{eqnarray*} \[\begin{eqnarray*} \end{eqnarray*}\], \[\begin{eqnarray*} Why do we need the \(I(\mbox{year=seniors})\) variable? Note that the opposite classifier to (H) might be quite good! \end{eqnarray*}\], \[\begin{eqnarray*} Multivariable logistic regression. biostat/vgsm/data/hersdata.txt, and it is described in Regression Methods in Biostatistics, page 30; variable descriptions are also given on the book website http://www.epibiostat.ucsf.edu/biostat/ vgsm/data/hersdata.codebook.txt. Consider the following data set collected from church offering plates in 62 consecutive Sundays. \end{eqnarray*}\] After adjusting for age, smoking is no longer significant. X_1 = \begin{cases} \[\begin{eqnarray*} \[\begin{eqnarray*} \end{eqnarray*}\], \[\begin{eqnarray*} Deep dive into Regression Analysis and how we can use this to infer mindboggling insights using Chicago COVID dataset. && \\ \end{eqnarray*}\] If classifier randomly guess, it should get half the positives correct and half the negatives correct. The Statistical Sleuth. With two consultants you might choose Sage first, and for the second option, it seems reasonable to choose the second most knowledgeable classmate (the second most highly associated variable), for example Bruno, who knows 75 topics. \mbox{sensitivity} &=& TPR = 300/308 = 0.974\\ 1995. The table below shows the result of the univariate analysis for some of the variables in the dataset. “Snoring as a Risk Factor for Disease: An Epidemiological Survey” 291: 630–32. 0 & \mbox{ survived} \[\begin{eqnarray*} augment contains the same number of rows as number of observations. Regression modeling of categorical or time-to-event outcomes with continuous and categorical predictors is covered. gamma: Goodman-Kruskal gamma is the number of concordant pairs minus the number of discordant pairs divided by the total number of pairs excluding ties. \end{eqnarray*}\], So, the LRT here is (see columns of null deviance and deviance): Heckman, and M.P. Given a particular pair, if the observation corresponding to a survivor has a higher probability of success than the observation corresponding to a death, we call the pair concordant. \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ OR &=& \mbox{odds dying if } (x_1, x_2) / \mbox{odds dying if } (x_1^*, x_2^*) = \frac{e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2}}{e^{\beta_0 + \beta_1 x_1^* + \beta_2 x_2^*}}\\ P(X=1 | p) &=& {4 \choose 1} p^1 (1-p)^{4-1}\\ p(x) &=& \beta_0 + \beta_1 x We can output the deviance ( = K - 2 * log-likelihood) for both the full (maximum likelihood!) \end{eqnarray*}\], \[\begin{eqnarray*} x &=& \mbox{log area burned} What about the RR (relative risk) or difference in risks? Instead, we’d like to predict new observations that were not used to create the model. &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ \mbox{specificity} &=& 92/127 = 0.724, \mbox{1 - specificity} = FPR = 0.276\\ That is, is the model able to discriminate between successes and failures.