regression methods in biostatistics datasets

\[\begin{eqnarray*} G &=& 2 \cdot \ln(L(MLE)) - 2 \cdot \ln(L(null))\\ Bayesian and Frequentist Regression Methods (Springer Series in Statistics) - Kindle edition by Wakefield, Jon. \end{eqnarray*}\], "~/Dropbox/teaching/math150/PracStatCD/Data Sets/Chapter 07/CSV Files/C7 Birdnest.csv", \[\begin{eqnarray*} &=& \bigg( \frac{1}{2 \pi \sigma^2} \bigg)^{n/2} e^{\sum_i (y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ The course will cover extensions of these methods to correlated data using generalized estimating equations. advantage of integrating multiple diverse datasets over analyzing them individually. However, (Menard 1995) warns that for large coefficients, standard error is inflated, lowering the Wald statistic (chi-square) value. Consider a toy example describing, for example, flipping coins. &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ Write out a few models by hand, does any of the significance change with respect to interaction? However, we may miss out of variables that are good predictors but aren’t linearly related. For many students and researchers learning to use these methods, this one book may be all they need to conduct and interpret multipredictor regression analyses. The first type of method applied logistic regression model with the four penalties to the merged data directly. 1 & \text{for always} \\ In general, the method of least squares is applied to obtain the equation of the regression line. \mathrm{logit}(p(x)) &=& \beta_0 + \beta_1 x\\ Graduate Prerequisites: The biostatistics and epidemiology MPH core course requirements and BS723 or BS852. \end{eqnarray*}\], \[\begin{eqnarray*} As we’ve seen, correlated variables cause trouble because they inflate the variance of the coefficient estimates. data described in Breslow and Day (1980) from a matched case control study. However, looking at all possible interactions (if only 2-way interactions, we could also consider 3-way interactions etc. \[\begin{eqnarray*} Unfortunately, you get carried away and spend all your time on memorizing the model answers to all past questions. &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ \[\begin{eqnarray*} Imagine you are preparing for your statistics exam. 1 - p(x) = \frac{1}{1+e^{\beta_0 + \beta_1 x}} OR &=& \mbox{odds dying if } (x_1, x_2) / \mbox{odds dying if } (x_1^*, x_2^*) = \frac{e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2}}{e^{\beta_0 + \beta_1 x_1^* + \beta_2 x_2^*}}\\ What about the RR (relative risk) or difference in risks? Note 2: We can see that smoking becomes less significant as we add age into the model. But if the new exam asks different questions about the same material, you would be ill-prepared and get a much lower mark than with a more traditional preparation. \end{eqnarray*}\]. \[\begin{eqnarray*} \[\begin{eqnarray*} 1985. Generally, extraneous variables are not so problematic because they produce models with unbiased coefficient estimators, unbiased predictions, and unbiased variance estimates. \hat{p}(2) &=& 0.7996326\\ x &=& \mbox{log area burned} \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ \end{cases} \end{eqnarray*}\], \[\begin{eqnarray*} Note 4 Every type of generalized linear model has a link function. (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 … 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}\\ These new methods can be used to perform prediction, estimation, and inference in complex big-data settings. We cannot reject the null hypothesis, so we know that we don’t need the weight in the model either. H_1: && \beta_1 \ne 0\\ x_2 &=& \begin{cases} &=& \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}}}\\ ), things can get out of hand quickly. i Fitting Regression Lines—The Method of Least Squares 2( )( ) 0 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. \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)}}}\\ P( \chi^2_1 \geq 5.11) &=& 0.0238 p(k) &=& 1-(1-\lambda)^k\\ Agresti, A. 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). where we are modeling the probability of 20-year mortality using smoking status and age group. Suppose that we build a classifier (logistic regression model) on a given data set. As done previously, we can add and remove variables based on the deviance. \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ Model building is definitely an ``art." How is it interpreted? The above inequality holds because \(\hat{\underline{p}}\) maximizes the likelihood. Heckman, and M.P. Another strategy for model building. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. book series S-curves ( y = exp(linear) / (1+exp(linear)) ) for a variety of different parameter settings. Maximum likelihood estimates are functions of sample data that are derived by finding the value of \(p\) that maximizes the likelihood functions. Some advanced topics are covered but the presentation remains intuitive. \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 \hat{p}(2.5) &=& 0.01894664\\ &=& -2 \Bigg[ \ln \bigg( (0.25)^{y} (0.75)^{n-y} \bigg) - \ln \Bigg( \bigg( \frac{y}{n} \bigg)^{y} \bigg( \frac{(n-y)}{n} \bigg)^{n-y} \Bigg) \Bigg]\\ \[\begin{eqnarray*} &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ 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. \[\begin{eqnarray*} H: is worse than random guessing. Sage. \end{eqnarray*}\] \mbox{test stat} &=& \chi^2\\ How do you choose the \(\alpha\) values? \end{eqnarray*}\], \[\begin{eqnarray*} \beta_{0s} &=& \beta_0 + \beta_2\\ 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. It turns out that we’ve also maximized the normal likelihood. No, you would guess \(p=0.25\)… you maximized the likelihood of seeing your data. Download it once and read it on your Kindle device, PC, phones or tablets. For simplicity, consider only first year students and seniors. This method of estimating the parameters of a regression line is known as the method of least squares. G &=& 525.39 - 335.23 = 190.16\\ Cengage Learning. \end{eqnarray*}\], \[\begin{eqnarray*} In particular, methods are illustrated using a variety of data sets. 1 & \text{for often} \\ 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. One idea is to start with an empty model and adding the best available variable at each iteration, checking for needs for transformations. 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). \[\begin{eqnarray*} Unsurprisingly, there are many approaches to model building, but here is one strategy, consisting of seven steps, that is commonly used when building a regression model. \end{eqnarray*}\]. Regression modeling of categorical or time-to-event outcomes with continuous and categorical predictors is covered. \end{eqnarray*}\] Use linear regression for prediction; Estimate the mean squared error of a predictive model; Use knn regression and knn classifier; Use logistic regression as a classification algorithm; Calculate the confusion matrix and evaluate the classification ability; Implement linear and quadratic discriminant … To assess a model’s accuracy (model assessment). \[\begin{eqnarray*} These methods, however, are not optimized for microbiome datasets. \mbox{additive model} &&\\ This course provides basic knowledge of logistic regression and analysis of survival data. The rules, however, state that you can bring two classmates as consultants. \end{eqnarray*}\] B: Let’s say we use prob=0.7 as a cutoff: \[\begin{eqnarray*} The authors cover t-tests, ANOVA and regression models, but also the advanced methods of generalised linear models and classification and regression … \mbox{specificity} &=& 92/127 = 0.724, \mbox{1 - specificity} = FPR = 0.276\\ 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. 1995. With logistic regression, we don’t have residuals, so we don’t have a value like \(R^2\). &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 P(X=1 | p = 0.05) &=& 0.171\\ \(e^{\beta_1}\) is the odds ratio for dying associated with a one unit increase in x. \mbox{& a loglikelihood of}: &&\\ Recall: x_2 &=& \begin{cases} Use stepwise regression, which of course only yields one model unless different alpha-to-remove and alpha-to-enter values are specified. What does it mean that the interaction terms are not significant in the last model? \[\begin{eqnarray*} (Agresti 1996) states that the likelihood-ratio test is more reliable for small sample sizes than the Wald test. &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ C: Let’s say we use prob=0.9 as a cutoff: \[\begin{eqnarray*} GLM: g(E[Y | X]) = \beta_0 + \beta_1 X 0 & \text{otherwise} \\ The results of HERS are surprising in light of previous observational studies, which found lower rates of CHD in women who take postmenopausal estrogen. That is, is the model able to discriminate between successes and failures. Part of Springer Nature. \(\beta_0\) now determines the location (median survival). &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) 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) \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ \end{eqnarray*}\] The logistic regression model is correct! 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. 1 & \mbox{ smoke}\\ Some intuition of both calculus and Linear Algebra will make your journey easier. \mbox{sensitivity} &=& TPR = 265/308 = 0.860\\ What does that even mean? 1996. \[\begin{eqnarray*} We will use \end{eqnarray*}\], \[\begin{eqnarray*} Generally: the idea is to use a model building strategy with some criteria (\(\chi^2\)-tests, AIC, BIC, ROC, AUC) to find the middle ground between an underspecified model and an overspecified model. The table below shows the result of the univariate analysis for some of the variables in the dataset. It gives you a sense of whether or not you’ve overfit the model in the building process.) the negative-binomial regression model in DESeq2 (Love and others, 2014) and overdispersed Poisson model in edgeR (Robinson and others, 2010). \end{eqnarray*}\], \[\begin{eqnarray*} sensitivity = power = true positive rate (TPR) = TP / P = TP / (TP+FN), false positive rate (FPR) = FP / N = FP / (FP + TN), positive predictive value (PPV) = precision = TP / (TP + FP), negative predictive value (NPV) = TN / (TN + FN), false discovery rate = 1 - PPV = FP / (FP + TP), one training set, one test set [two drawbacks: estimate of error is highly variable because it depends on which points go into the training set; and because the training data set is smaller than the full data set, the error rate is biased in such a way that it overestimates the actual error rate of the modeling technique. \end{cases} Suppose also that you know which topics each of your classmates is familiar with. (see log-linear model below, ). &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ P(X=1 | p = 0.25) &=& 0.422\\ \end{eqnarray*}\], \[\begin{eqnarray*} If you set \(\alpha_e\) to be very small, you might walk away with no variables in your model, or at least not many. The logistic regression model is overspecified. By using Kaggle, you agree to our use of cookies. \end{eqnarray*}\], \[\begin{eqnarray*} A study was undertaken to investigate whether snoring is related to a heart disease. \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) \end{eqnarray*}\], \[\begin{eqnarray*} Randomly divide the data into a training set and a validation set: Using the training set, identify several candidate models: And, most of all, don’t forget that there is not necessarily only one good model for a given set of data. \mbox{specificity} &=& 120/127 = 0.945, \mbox{1 - specificity} = FPR = 0.055\\ If the observation corresponding to a survivor has a lower probability of success than the observation corresponding to a death, we call the pair discordant. John Wiley; Sons, New York. \end{eqnarray*}\], Using the logistic regression model makes the likelihood substantially more complicated because the probability of success changes for each individual. A Receiver Operating Characteristic (ROC) Curve is a graphical representation of the relationship between. The Statistical Sleuth. \end{cases}\\ Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt \[\begin{eqnarray*} We can see that the logit transformation linearizes the relationship. The second type is MetaLasso, and our proposed method is as the third type. The output generated differs slightly from that shown in the tables. A large cross-validation AUC on the validation data is indicative of a good predictive model (for your population of interest). \mathrm{logit}(p) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 Consider the following data set collected from church offering plates in 62 consecutive Sundays. Biostatistical Methods Overview, Programs and Datasets (First Edition) ... fits the Poisson regression models using the SAS program shown in Table 8.2 that generates the output shown in Tables 8.3, 8.4 and 8.5. Taken from For inferential reasons - that is, the model will be used to explore the strength of the relationships between the response and the predictors. We can use the drop-in-deviance test to test the effect of any or all of the parameters (of which there are now four) in the model. -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 \hat{p} &=& \frac{49}{147}\\ After adjusting for age, smoking is no longer significant. E[\mbox{grade seniors}| \mbox{hours studied}] &=& \beta_{0s} + \beta_{1s} \mbox{hrs}\\ (The fourth step is very good modeling practice. Example 5.3 Consider the example on smoking and 20-year mortality (case) from section 3.4 of Regression Methods in Biostatistics, pg 52-53. \end{eqnarray*}\], \[\begin{eqnarray*} Cancer Linear Regression. AIC: Akaike’s Information Criteria = \(-2 \ln\) likelihood + \(2p\) p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 They also show that these regression methods deal with confounding, mediation, and interaction of causal effects in essentially the same way. y &=& \begin{cases} Datasets Most of the datasets on this page are in the S dumpdata and R compressed save() file formats. \end{eqnarray*}\], \[\begin{eqnarray*} But more importantly, age is a variable that reverses the effect of smoking on cancer - Simpson’s Paradox. Over 10 million scientific documents at your fingertips. \end{eqnarray*}\]. Recall, when comparing two nested models, the differences in the deviances can be modeled by a \(\chi^2_\nu\) variable where \(\nu = \Delta p\). \[\begin{eqnarray*} Note 3 && \\ \end{eqnarray*}\] Not logged in It won’t be constant for a given \(X\), so it must be calculated as a function of \(X\). \end{eqnarray*}\], \[\begin{eqnarray*} A short summary of the book is provided elsewhere, on a short post (Feb. 2008). \[\begin{eqnarray*} -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu Instead, we’d like to predict new observations that were not used to create the model. We are going to discuss how to add (or subtract) variables from a model. Example 5.4 Suppose that you have to take an exam that covers 100 different topics, and you do not know any of them. Is a different picture provided by considering odds? The functional form relating x and the probability of success looks like it could be an S shape. X_1 = \begin{cases} \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x In particular, methods are illustrated using a variety of data sets. Consider false positive rate, false negative rate, outliers, parsimony, relevance, and ease of measurement of predictors. \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) Recall that logistic regression can be used to predict the outcome of a binary event (your response variable). The examples, analyzed using Stata, are drawn from the biomedical context but generalize to other areas of application. Applied Logistic Regression Analysis. With correlated variables it is still possible to get unbiased prediction estimates, but the coefficients themselves are so variable that they cannot be interpreted (nor can inference be easily performed). We can output the deviance ( = K - 2 * log-likelihood) for both the full (maximum likelihood!) \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ \end{eqnarray*}\] p(x) &=& 1 - \exp [ -\exp(\beta_0 + \beta_1 x) ] where \(\nu\) is the number of extra parameters we estimate using the unconstrained likelihood (as compared to the constrained null likelihood). Ramsey, F., and D. Schafer. \mbox{simple model} &&\\ If we are testing only one parameter value. Evaluate the selected models for violation of the model conditions. G: random guessing. p-value &=& P(\chi^2_1 \geq 190.16) = 0 p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ 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. In the burn data we have 308 survivors and 127 deaths = 39,116 pairs of people. Using the burn data, convince yourself that the RR isn’t constant. This dataset includes data taken from about deaths due to cancer in the United States. \end{eqnarray*}\], So, the LRT here is (see columns of null deviance and deviance): p_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} 3rd ed. P(X=1 | p = 0.05) &=& 0.171\\ e^{0} &=& 1\\ \mathrm{logit}(p(x)) &=& \beta_0 + \beta_1 x\\ \end{eqnarray*}\]. 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. Just like in linear regression, our Y response is the only random component. p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} Maximizing the likelihood? p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ \end{eqnarray*}\], \[\begin{eqnarray*} p(-\beta_0 / \beta_1) &=& p(x) = 0.5 A: Let’s say we use prob=0.25 as a cutoff: \[\begin{eqnarray*} There might be a few equally satisfactory models. \mathrm{logit}(p(x+1)) &=& \beta_0 + \beta_1 (x+1)\\ augment contains the same number of rows as number of observations. \end{eqnarray*}\], \[\begin{eqnarray*} 0 & \mbox{ don't smoke}\\ &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ 2 Several methods that remove or adjust batch variation have been developed. &=& -2 [ \ln(0.0054) - \ln(0.0697) ] = 5.11\\ &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ Continue removing variables until all variables are significant at the chosen. p(x) &=& \beta_0 + \beta_1 x © 2020 Springer Nature Switzerland AG. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ The hormone replacement regimen also increased the risk of clots in the veins (deep vein thrombosis) and lungs (pulmonary embolism). Biostatistics with R provides a straightforward introduction on how to analyse data from the wide field of biological research, including nature protection and global change monitoring. What we see is that the vast majority of the controls were young, and they had a high rate of smoking. \end{eqnarray*}\] In fact, usually, we use them to test whether the coefficients are zero: \[\begin{eqnarray*} You begin by trying to answer the questions from previous papers and comparing your answers with the model answers provided. \mbox{test stat} &=& \chi^2\\ H_0:&& p=0.25\\ L(\underline{p}) &=& \prod_i \Bigg( \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)^{y_i} \Bigg(1-\frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)^{(1- y_i)} \\ \end{eqnarray*}\], \[\begin{eqnarray*} This method of estimating the parameters of a regression line is known as the method of least squares. G &=& 3597.3 - 3594.8 =2.5\\ which gives a likelihood of: In machine learning, these methods are known as regression (for continuous outcomes) and classification (for categorical outcomes) methods. WHY??? [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). ] \end{eqnarray*}\], \[\begin{eqnarray*} For logistic regression, we use the logit link function: \[\begin{eqnarray*} H_1: && \beta_1 \ne 0\\ Norton, P.G., and E.V. gamma: Goodman-Kruskal gamma is the number of concordant pairs minus the number of discordant pairs divided by the total number of pairs excluding ties. \end{eqnarray*}\], \[\begin{eqnarray*} -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu This new book provides a unified, in-depth, readable introduction to the multipredictor regression methods most widely used in biostatistics: linear models for continuous outcomes, logistic models for binary outcomes, the Cox model for right-censored survival times, repeated-measures models for longitudinal and hierarchical outcomes, and generalized linear models for counts and other outcomes. During investigation of the US space shuttle Challenger disaster, it was learned that project managers had judged the probability of mission failure to be 0.00001, whereas engineers working on the project had estimated failure probability at 0.005. \beta_{1s} &=& \beta_1 + \beta_3 \end{eqnarray*}\], When each person is at risk for a different covariate (i.e., explanatory variable), they each end up with a different probability of success. Robust Methods in Biostatistics proposes robust alternatives to common methods used in statistics in general and in biostatistics in particular and illustrates their use on many biomedical datasets. 0 & \mbox{ don't smoke}\\ Therefore, if its possible, a scatter plot matrix would be best. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. 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})\\ &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 Age seems to be less important than drinking status. We will focus here only on model assessment. p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087\\ Recall that the response variable is binary and represents whether there is a small opening (closed=1) or a large opening (closed=0) for the nest. The link is the relationship between the response variable and the linear function in x. \mbox{old} & \mbox{65+ years old}\\ The odds ratio \(\hat{OR}_{1.90, 2.00}\) is given by The logistic regression model is underspecified. \mbox{middle} & \mbox{45-64 years old}\\ The validation set is used for cross-validation of the fitted model. Also noted is whether there was enough change to buy a candy bar for $1.25. Statistics for Biology and Health Bayesian and Frequentist Regression Methods Website. 1. Once \(y_1, y_2, \ldots, y_n\) have been observed, they are fixed values. 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 … The pairs would be concordant if the first individual survived and the second didn’t. uses the Mack et al. \end{eqnarray*}\] && \\ \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ \beta_{1f} &=& \beta_1\\ \[\begin{eqnarray*} \[\begin{eqnarray*} \ln[ - \ln (1-p(k))] &=& \beta_0 + 1 \cdot \ln(k)\\ \end{eqnarray*}\], 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!) \end{eqnarray*}\]. More on this as we move through this model. -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) &=& -2 [ \ln (L(p_0)) - \ln(L(\hat{p}))]\\ P(X=1 | p = 0.9) &=& 0.0036 \\ Cross validation is commonly used to perform two different tasks: 2012. We can show that if \(H_0\) is true, 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). [\(\beta_1\) is the change in log-odds associated with a one unit increase in x. 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. p(0) = \frac{e^{\beta_0}}{1+e^{\beta_0}} 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. \end{eqnarray*}\], \[\begin{eqnarray*} RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ \end{cases} \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ To account for the variation in sequencing depth and high dimensionality of read counts, a high-dimensional log-contrast model is often used where log compositions of read counts are used as covariates. \end{eqnarray*}\] \beta_{0f} &=& \beta_{0}\\ \mathrm{logit}(\hat{p}) = 22.708 - 10.662 \cdot \ln(\mbox{ area }+1). 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. where \(y_1, y_2, \ldots, y_n\) represents a particular observed series of 0 or 1 outcomes and \(p\) is a probability \(0 \leq p \leq 1\). \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)\\ While a first course in statistics is assumed, a chapter reviewing basic statistical methods is included. \[\begin{eqnarray*} The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. Example 4.3 Consider a simple linear regression model on number of hours studied and exam grade. An Introduction to Categorical Data Analysis. The logistic regression model is a generalizedlinear model. The majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. But we’d have to do some work to figure out what the form of that S looks like. Consider the HERS data described in your book (page 30); variable description also given on the book website BIOST 570 Advanced Regression Methods for Independent Data (3) Covers linear models, generalized linear and non-linear regression, and models. Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt, 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. p-value &=& P(\chi^2_6 \geq 9.1)= 1 - pchisq(9.1, 6) = 0.1680318 \[\begin{eqnarray*} \end{eqnarray*}\] 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 to make a request. \[\begin{eqnarray*} That is, the difference in log likelihoods will be the opposite difference in deviances: 1. Length as a continuous explanatory variable: Length as a categorical explanatory variables: Length plus a few other explanatory variables: \[\begin{eqnarray*} The likelihood is the probability distribution of the data given specific values of the unknown parameters. Co-organized by the Department of Biostatistics at the Harvard T.H. Lesson of the story: be very very very careful interpreting coefficients when you have multiple explanatory variables. We will study Linear Regression, Polynomial Regression, Normal equation, gradient descent and step by step python implementation. \mbox{overall OR} &=& e^{-0.37858 } = 0.6848332\\ 0 &=& (1-p) \sum_i y_i + p (n-\sum_i y_i) \\ \hat{p}(1.5) &=& 0.9987889\\ p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} 1 & \text{for occasionally} \\ \end{eqnarray*}\], \[\begin{eqnarray*} Y_i \sim \mbox{Bernoulli} \bigg( p(x_i) = \frac{e^{\beta_0 + \beta_1 x_i}}{1+ e^{\beta_0 + \beta_1 x_i}}\bigg) \end{cases} G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} Advanced Methods in Biostatistics IV - Regression Modeling Advanced Methods in Biostatistics IV covers topics in modern multivariate regression from estimation theoretic, likelihood-based, and Bayesian points of view. We can now model binary response variables. \end{eqnarray*}\]. \mbox{sensitivity} &=& TPR = 144/308 = 0.467\\ Study bivariate relationships to reveal other outliers, to suggest possible transformations, and to identify possible multicollinearities. \mbox{& a loglikelihood of}: &&\\ to log(area +1)= 2.00. I can’t possibly over-emphasize the data exploration step. We require that \(\alpha_e<\alpha_l\), otherwise, our algorithm could cycle, we add a variable, then immediately decide to delete it, continuing ad infinitum. \hat{RR}_{1, 2} &=& 1.250567\\ \end{cases} \end{eqnarray*}\]. 0 & \text{otherwise} \\ 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. \[\begin{eqnarray*} Hulley, S., D. Grady, T. Bush, C. Furberg, D. Herrington, B. Riggs, and E. Vittinghoff. x_1 &=& \begin{cases} gives the \(\ln\) odds of success . There’s not a data analyst out there who hasn’t made the mistake of skipping this step and later regretting it when a data point was found in error, thereby nullifying hours of work. 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 & \text{otherwise} \\ The datasets below will be used throughout this course. 1 & \mbox{ died}\\ The estimates have an approximately normal sampling distribution for large sample sizes because they are maximum likelihood estimates. Another worry when building models with multiple explanatory variables has to do with variables interacting. \end{eqnarray*}\] 0 & \mbox{ survived} We can now model binary response variables. \end{cases}\\ Each woman was randomly assigned to receive one tablet containing 0.625 mg conjugated estrogens plus 2.5 mg medroxyprogesterone acetate daily or an identical placebo. We can, however, measure whether or not the estimated model is consistent with the data. That is, the variables contain the same information as other variables (i.e., are correlated!). \end{eqnarray*}\], \[\begin{eqnarray*} where \(\hat{\beta}_0\) is fit from a model without any explanatory variable, \(x\). The participants are postmenopausal women with a uterus and with CHD. P(Y=y) &=& p^y(1-p)^{1-y} Not affiliated &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ The summary contains the following elements: number of observations used in the fit, maximum absolute value of first derivative of log likelihood, model likelihood ratio chi2, d.f., P-value, \(c\) index (area under ROC curve), Somers’ Dxy, Goodman-Kruskal gamma, Kendall’s tau-a rank correlations between predicted probabilities and observed response, the Nagelkerke \(R^2\) index, the Brier score computed with respect to Y \(>\) its lowest level, the \(g\)-index, \(gr\) (the \(g\)-index on the odds ratio scale), and \(gp\) (the \(g\)-index on the probability scale using the same cutoff used for the Brier score). Note that the x-axis is some continuous variable x while the y-axis is the probability of success at that value of x. In the survey, 2484 people were classified according to their proneness to snoring (never, occasionally, often, always) and whether or not they had the heart disease. &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ This service is more advanced with JavaScript available, Part of the Let’s say this is Sage who knows 85 topics. \hat{p} &=& \frac{ \sum_i y_i}{n} In a broader sense, the merging of several datasets into one single dataset also constitutes a batch effect problem. \[\begin{eqnarray*} \end{eqnarray*}\], Our new model becomes: \end{eqnarray*}\] “Snoring as a Risk Factor for Disease: An Epidemiological Survey” 291: 630–32. Consider looking at all the pairs of successes and failures. && \\ p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \mbox{old} & \mbox{65+ years old}\\ \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ \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 and reduced (null) models. \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ Multivariable logistic regression. \ln[ - \ln (1-p(k))] &=& \ln[-\ln(1-\lambda)] + \ln(k)\\ 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 . \mathrm{logit}(\star) = \ln \bigg( \frac{\star}{1-\star} \bigg) \ \ \ \ 0 < \star < 1 2nd ed. If classifier randomly guess, it should get half the positives correct and half the negatives correct. x &=& - \beta_0 / \beta_1\\ That is, the variables are important in predicting odds of survival. 1998. \end{eqnarray*}\], \[\begin{eqnarray*} However, within each group, the cases were more likely to smoke than the controls. \end{eqnarray*}\], \[\begin{eqnarray*} \hat{p} &=& \frac{ \sum_i y_i}{n} “Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions.” Journal of the American Statistical Association, 141–50. Statistical tools for analyzing experiments involving genomic data. \end{eqnarray*}\]. For now, we will try to predict whether the individuals had a medical condition, medcond. \end{eqnarray*}\], \[\begin{eqnarray*} 5, 6 Undetected batch effects can have major impact on subsequent conclusions in both unsupervised and supervised analysis. The first half of the course introduces descriptive statistics and the inferential statistical methods of confidence intervals and significance tests. We can estimate the SE (Wald estimates via Fisher Information). “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. Where \(p(x)\) is the probability of success (here surviving a burn). We do have good reasons for how we defined it, but that doesn’t mean there aren’t other good ways to model the relationship.). 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). To maximize the likelihood, we use the natural log of the likelihood (because we know we’ll get the same answer): The second half introduces bivariate and multivariate methods, emphasizing contingency table analysis, regression, and analysis of variance. Regression Methods in Biostatistics This page contains R scripts for doing the analysis presented in the book entitled Regression Methods in Biostatistics (Eric Vittinghoff, David V. Glidden, Stephen C. Shiboski, and Charles E. McCulloch, Springer 2005). Note that tidy contains the same number of rows as the number of coefficients. \end{eqnarray*}\], \[\begin{eqnarray*} \end{eqnarray*}\], \[\begin{eqnarray*} E[\mbox{grade first years}| \mbox{hours studied}] &=& \beta_{0f} + \beta_{1f} \mbox{hrs}\\ \mbox{test stat} &=& G\\ Supplemented with numerous graphs, charts, and tables as well as a Web site for larger data sets and exercises, Biostatistical Methods: The Assessment of Relative Risks is an excellent guide for graduate-level students in biostatistics and an invaluable reference for biostatisticians, applied statisticians, and epidemiologists. \end{eqnarray*}\], Example 5.1 Surviving third-degree burns 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. \end{eqnarray*}\] On a univariate basis, check for outliers, gross data errors, and missing values.

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