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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. Graduate Prerequisites: The biostatistics and epidemiology MPH core course requirements and BS723 or BS852. 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. Another worry when building models with multiple explanatory variables has to do with variables interacting. 2. \end{eqnarray*}\], $\begin{eqnarray*} They also show that these regression methods deal with confounding, mediation, and interaction of causal effects in essentially the same way. p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 Try computing the RR at 1.5 versus 2.5, then again at 1 versus 2. Cengage Learning. Unfortunately, you get carried away and spend all your time on memorizing the model answers to all past questions. These methods, however, are not optimized for microbiome datasets. augment contains the same number of rows as number of observations. Dunn. \ln L(p) &=& \ln \Bigg(p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)} \Bigg)\\ &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ G &=& 3597.3 - 3594.8 =2.5\\ That is, the variables contain the same information as other variables (i.e., are correlated!). The pairs would be discordant if the first individual died and the second survived. \end{eqnarray*}$, $\begin{eqnarray*} The logistic regression model is correct! For inferential reasons - that is, the model will be used to explore the strength of the relationships between the response and the predictors. For logistic regression, we use the logit link function: We should also look at interactions which we might suspect. S-curves ( y = exp(linear) / (1+exp(linear)) ) for a variety of different parameter settings. X_3 = \begin{cases} In general, there are five reasons one might want to build a regression model. (Fan, Heckman, and Wand 1995). The effect is not due to the observational nature of the study, and so it is important to adjust for possible influential variables regardless of the study at hand. \end{eqnarray*}$, $\begin{eqnarray*} and reduced (null) models. Suppose that we build a classifier (logistic regression model) on a given data set. \end{eqnarray*}$, $\begin{eqnarray*} During investigation of the US space shuttle Challenger disaster, it was learned that project managers had judged the probability of mission failure to be 0.00001, whereas engineers working on the project had estimated failure probability at 0.005. Use linear regression for prediction; Estimate the mean squared error of a predictive model; Use knn regression and knn classifier; Use logistic regression as a classification algorithm; Calculate the confusion matrix and evaluate the classification ability; Implement linear and quadratic discriminant … 198.71.239.51, applied regression methods for biomedical research, linear, logistic, generalized linear, survival (Cox), GEE, a, Department of Epidemiology and Biostatistics, Springer Science+Business Media, Inc. 2005, Repeated Measures and Longitudinal Data Analysis. \mathrm{logit}(\hat{p}) &=& 22.708 - 10.662 \cdot \ln(\mbox{ area }+1)\\ x_2 &=& \begin{cases} &=& \bigg( \frac{1}{2 \pi \sigma^2} \bigg)^{n/2} e^{\sum_i (y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ What does it mean that the interaction terms are not significant in the last model? \end{eqnarray*}$ (Think about Simpson’s Paradox and the need for interaction.). The hormone replacement regimen also increased the risk of clots in the veins (deep vein thrombosis) and lungs (pulmonary embolism). P(X=1 | p = 0.05) &=& 0.171\\ We will use the variables age, weight, diabetes and drinkany. always. \frac{p(x)}{1-p(x)} = e^{\beta_0 + \beta_1 x} P(X=1 | p = 0.9) &=& 0.0036 \\ \end{eqnarray*}\], $\begin{eqnarray*} A large cross-validation AUC on the validation data is indicative of a good predictive model (for your population of interest). We can now model binary response variables. \[\begin{eqnarray*} To assess a model’s accuracy (model assessment). After adjusting for age, smoking is no longer significant. \hat{p}(1.5) &=& 0.9987889\\ H_0: && \beta_1 =0\\ \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ \end{eqnarray*}$, $\begin{eqnarray*} which gives a likelihood of: -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 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. (The fourth step is very good modeling practice. Cancer Linear Regression. gives the $$\ln$$ odds of success . This service is more advanced with JavaScript available, Part of the \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$ Once $$y_1, y_2, \ldots, y_n$$ have been observed, they are fixed values. \end{eqnarray*}\], $\begin{eqnarray*} \mathrm{logit}(p(x+1)) &=& \beta_0 + \beta_1 (x+1)\\ 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. But more importantly, age is a variable that reverses the effect of smoking on cancer - Simpson’s Paradox. Start with the full model including every term (and possibly every interaction, etc.). \mbox{& a loglikelihood of}: &&\\ However, looking at all possible interactions (if only 2-way interactions, we could also consider 3-way interactions etc. \[\begin{eqnarray*} 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}\\ A study was undertaken to investigate whether snoring is related to a heart disease. \end{eqnarray*}$, Using the logistic regression model makes the likelihood substantially more complicated because the probability of success changes for each individual. $\begin{eqnarray*} The first type of method applied logistic regression model with the four penalties to the merged data directly. \[\begin{eqnarray*} Collect the data. In a broader sense, the merging of several datasets into one single dataset also constitutes a batch effect problem. 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. That is, the difference in log likelihoods will be the opposite difference in deviances: 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. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ \end{cases} 0 & \text{otherwise} \\ p-value &=& P(\chi^2_6 \geq 9.1)= 1 - pchisq(9.1, 6) = 0.1680318 Introductory course in the analysis of Gaussian and categorical data. \hat{p}(1) &=& 0.9999941\\ 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. \[\begin{eqnarray*} \mbox{sensitivity} &=& TPR = 144/308 = 0.467\\ 1 - p(x) = \frac{1}{1+e^{\beta_0 + \beta_1 x}} \end{eqnarray*}$, $\begin{eqnarray*} Consider looking at all the pairs of successes and failures. The big model (with all of the interaction terms) has a deviance of 3585.7; the additive model has a deviance of 3594.8. Where $$p(x)$$ is the probability of success (here surviving a burn). Just like in linear regression, our Y response is the only random component. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. However, within each group, the cases were more likely to smoke than the controls. Linear Regression Datasets for Machine Learning. \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!) Use features like bookmarks, note taking and highlighting while reading Bayesian and Frequentist Regression Methods (Springer Series in Statistics). &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ P(X=1 | p = 0.15) &=& 0.368\\ The rules, however, state that you can bring two classmates as consultants. Some advanced topics are covered but the presentation remains intuitive. We start with the response variable versus all variables and find the best predictor. RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ 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)} \\ &=& \mbox{null (restricted) deviance - residual (full model) deviance}\\ p(x) &=& 1 - \exp [ -\exp(\beta_0 + \beta_1 x) ] 1 & \text{for often} \\ 1 - p(x) = \frac{1}{1+e^{\beta_0 + \beta_1 x}} \end{eqnarray*}\], $\begin{eqnarray*} &=& p^{y_1}(1-p)^{1-y_1} p^{y_2}(1-p)^{1-y_2} \cdots p^{y_n}(1-p)^{1-y_n}\\ \end{eqnarray*}$, $\begin{eqnarray*} Important note: \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ Note that the opposite classifier to (H) might be quite good! 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. 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. \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x 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. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. L(\hat{\underline{p}}) > L(p_0) GLM: g(E[Y | X]) = \beta_0 + \beta_1 X The estimates have an approximately normal sampling distribution for large sample sizes because they are maximum likelihood estimates. The datasets below will be used throughout this course. \[\begin{eqnarray*} Recall how we estimated the coefficients for linear regression. 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. 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). Datasets Most of the datasets on this page are in the S dumpdata and R compressed save() file formats. data described in Breslow and Day (1980) from a matched case control study. \[\begin{eqnarray*} BIOST 570 Advanced Regression Methods for Independent Data (3) Covers linear models, generalized linear and non-linear regression, and models. But we’d have to do some work to figure out what the form of that S looks like. This method of estimating the parameters of a regression line is known as the method of least squares. \end{cases}\\ \end{eqnarray*}$, $\begin{eqnarray*} Note 4 Every type of generalized linear model has a link function. Recall, when comparing two nested models, the differences in the deviances can be modeled by a $$\chi^2_\nu$$ variable where $$\nu = \Delta p$$. 1995. Heckman, and M.P. 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). We see above that the logistic model imposes a constant OR for any value of $$X$$ (and not a constant RR). 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. We can, however, measure whether or not the estimated model is consistent with the data. \end{cases} Introductory course in the analysis of Gaussian and categorical data. When using our method, we set μ=1 and α=0.5 except LASSO penalty. Would you guess $$p=0.49$$?? \end{eqnarray*}$. \mathrm{logit}(p) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 E[\mbox{grade first years}| \mbox{hours studied}] &=& \beta_{0f} + \beta_{1f} \mbox{hrs}\\ Using the burn data, convince yourself that the RR isn’t constant. \end{eqnarray*}\], $\begin{eqnarray*} \end{eqnarray*}$ Or, we can think about it as a set of independent binary responses, $$Y_1, Y_2, \ldots Y_n$$. Statistics for Biology and Health P(X=1 | p) &=& {4 \choose 1} p^1 (1-p)^{4-1}\\ We minimized the residual sum of squares: $\begin{eqnarray*} The methods introduced include robust estimation, testing, model selection, model check and diagnostics. This is designed to be a first course in Statistics. Taken from https://onlinecourses.science.psu.edu/stat501/node/332. \[\begin{eqnarray*} where $$g(\cdot)$$ is the link function. \[\begin{eqnarray*} \end{eqnarray*}$ y &=& \begin{cases} Therefore, if its possible, a scatter plot matrix would be best. Lesson of the story: be very very very careful interpreting coefficients when you have multiple explanatory variables. \end{eqnarray*}\] 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). &=& \bigg( \frac{1}{2 \pi \sigma^2} \bigg)^{n/2} e^{\sum_i (y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ $\begin{eqnarray*} L(\hat{\underline{p}}) > L(p_0) x_2 &=& \begin{cases} Contains notes on computations at the end of most chapters, covering the use of Excel, SAS, and others. 1 & \mbox{ died}\\ The pairs would be concordant if the first individual survived and the second didn’t. The above inequality holds because $$\hat{\underline{p}}$$ maximizes the likelihood. \hat{p}(2) &=& 0.7996326\\ 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. We applied three types of methods to these two datasets. &=& -2 [ \ln(0.0054) - \ln(0.0697) ] = 5.11\\ 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. \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ The link is the relationship between the response variable and the linear function in x. \[\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} For example: consider a pair of individuals with burn areas of 1.75 and 2.35. Wand. G &=& 525.39 - 335.23 = 190.16\\ &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. Ramsey, F., and D. Schafer. The logistic regression model is underspecified. The additive model has a deviance of 3594.8; the model without weight is 3597.3. where $$\nu$$ is the number of extra parameters we estimate using the unconstrained likelihood (as compared to the constrained null likelihood). Part of Springer Nature. C: Let’s say we use prob=0.9 as a cutoff: \[\begin{eqnarray*} Study bivariate relationships to reveal other outliers, to suggest possible transformations, and to identify possible multicollinearities. Age seems to be less important than drinking status. Using the additive model above: Note that the x-axis is some continuous variable x while the y-axis is the probability of success at that value of x. Regression modeling of categorical or time-to-event outcomes with continuous and categorical predictors is covered. \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_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} The validation set is used for cross-validation of the fitted model. \end{eqnarray*}$, $\begin{eqnarray*} H_0: && \beta_1 =0\\ &=& p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)}\\ Use stepwise regression, which of course only yields one model unless different alpha-to-remove and alpha-to-enter values are specified. p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 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^*}}\\ gives the probability of failure. WHY??? Consider a toy example describing, for example, flipping coins. &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 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). Statistical tools for analyzing experiments involving genomic data. \ln (p(x)) = \beta_0 + \beta_1 x Bayesian and Frequentist Regression Methods Website. In particular, methods are illustrated using a variety of data sets. \end{eqnarray*}$, $\begin{eqnarray*} RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ Tied pairs occur when the observed survivor has the same estimated probability as the observed death. The likelihood is the probability distribution of the data given specific values of the unknown parameters. p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ Fan, J., N.E. 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.). When a model is overspecified, there are one or more redundant variables. Also problematic is that the model becomes unnecessarily complicated and harder to interpret. Consider false positive rate, false negative rate, outliers, parsimony, relevance, and ease of measurement of predictors. Continue removing variables until all variables are significant at the chosen. \end{eqnarray*}$. \end{eqnarray*}\] The table below shows the result of the univariate analysis for some of the variables in the dataset. AIC: Akaike’s Information Criteria = $$-2 \ln$$ likelihood + $$2p$$ It turns out that we’ve also maximized the normal likelihood. \beta_0 + \beta_1 x &=& 0\\ 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)} \\ \ln[ - \ln (1-p(k))] &=& \ln[-\ln(1-\lambda)] + \ln(k)\\ (Agresti 1996) states that the likelihood-ratio test is more reliable for small sample sizes than the Wald test. &=& -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]\\ H_0:&& p=0.25\\ We are going to discuss how to add (or subtract) variables from a model. Note 2: We can see that smoking becomes less significant as we add age into the model. A: Let’s say we use prob=0.25 as a cutoff: $\begin{eqnarray*} \beta_{0f} &=& \beta_{0}\\ The book is centred around traditional statistical approaches, focusing on those prevailing in research publications. \end{cases} &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 \mbox{old} & \mbox{65+ years old}\\ P(X=1 | p = 0.25) &=& 0.422\\ \mbox{overall OR} &=& e^{-0.37858 } = 0.6848332\\ \beta_{0s} &=& \beta_0 + \beta_2\\ \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ How is it interpreted? \mbox{test stat} &=& \chi^2\\ 2012. They are: Decide which explanatory variables and response variable on which to collect the data. For control purposes - that is, the model will be used to control a response variable by manipulating the values of the predictor variables. P(X=1 | p = 0.05) &=& 0.171\\ \end{eqnarray*}$ Next we will check whether we need weight. It won’t be constant for a given $$X$$, so it must be calculated as a function of $$X$$. Recall: Being underspecified is the worst case scenario because the model ends up being biased and predictions are wrong for virtually every observation. What we see is that the vast majority of the controls were young, and they had a high rate of smoking. $\begin{eqnarray*} \mbox{test stat} &=& \chi^2\\ An Introduction to Categorical Data Analysis. Because we will use maximum likelihood parameter estimates, we can also use large sample theory to find the SEs and consider the estimates to have normal distributions (for large sample sizes). P(X=1 | p = 0.75) &=& 0.047 \\ \end{eqnarray*}$ We continue with this process until there are no more variables that meet either requirements. \mbox{specificity} &=& 92/127 = 0.724, \mbox{1 - specificity} = FPR = 0.276\\ &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ \ln[ - \ln (1-p(k))] &=& \beta_0 + 1 \cdot \ln(k)\\ \mathrm{logit}(\hat{p}) &=& 22.708 - 10.662 \cdot \ln(\mbox{ area }+1)\\ e^{\beta_1} &=& \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ Is a different picture provided by considering odds? &=& -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]\\ Note 3 &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ Note 1: We can see from above that the coefficients for each variable are significantly different from zero. The course will cover extensions of these methods to correlated data using generalized estimating equations. [$$\beta_1$$ is the change in log-odds associated with a one unit increase in x. \hat{RR}_{1.5, 2.5} &=& 52.71587\\ The difference between these two probabilities, 0.00499 was discounted as being too small to worry about. MackLogi.sas: uses the Mack et al. P(X=1 | p = 0.5) &=& 0.25\\ ), things can get out of hand quickly. It gives you a sense of whether or not you’ve overfit the model in the building process.) Now, if the upcoming exam completely consists of past questions, you are certain to do very well. We will study Linear Regression, Polynomial Regression, Normal equation, gradient descent and step by step python implementation. We can see that the logit transformation linearizes the relationship. In particular, methods are illustrated using a variety of data sets. However, we may miss out of variables that are good predictors but aren’t linearly related. $\begin{eqnarray*} x &=& - \beta_0 / \beta_1\\ (see log-linear model below, 5.1.2.1 ). X_2 = \begin{cases} \end{eqnarray*}$ \end{eqnarray*}\]. \end{eqnarray*}\] For predictive reasons - that is, the model will be used to predict the response variable from a chosen set of predictors. $\begin{eqnarray*} The odds ratio $$\hat{OR}_{1.90, 2.00}$$ is given by $$\beta_0$$ now determines the location (median survival). Let’s say $$X \sim Bin(p, n=4).$$ We have 4 trials and $$X=1$$. &=& \ln \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ \mbox{simple model} &&\\ p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} In many situations, this will help us from stopping at a less than desirable model. \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) \[\begin{eqnarray*} Not affiliated 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). That is, the variables are important in predicting odds of survival. && \\ p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 Below I’ve given some different relationships between x and the probability of success using $$\beta_0$$ and $$\beta_1$$ values that are yet to be defined. Why do we need the $$I(\mbox{year=seniors})$$ variable? \end{eqnarray*}$. $\begin{eqnarray*} \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ We can now model binary response variables. \[\begin{eqnarray*} However, (Menard 1995) warns that for large coefficients, standard error is inflated, lowering the Wald statistic (chi-square) value. Likelihood? \end{cases} \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ It seems that a transformation of the data is in place. Norton, P.G., and E.V. \[\begin{eqnarray*} P(X=1 | p = 0.5) &=& 0.25\\ Though it is important to realize that we cannot find estimates in closed form. 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. p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} 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. 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. -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu 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)\\ 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. \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) 1 & \mbox{ smoke}\\ Multivariable logistic regression. 0 & \mbox{ don't smoke}\\ &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ book series These data refer to 435 adults who were treated for third-degree burns by the University of Southern California General Hospital Burn Center. \mbox{test stat} &=& G\\ -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu Before reading the notes here, look through the following visualization. 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. Model building is definitely an art." One idea is to start with an empty model and adding the best available variable at each iteration, checking for needs for transformations. B: Let’s say we use prob=0.7 as a cutoff: \[\begin{eqnarray*} P( \chi^2_1 \geq 5.11) &=& 0.0238 where we are modeling the probability of 20-year mortality using smoking status and age group. That is because age and smoking status are so highly associated (think of the coin example). A short summary of the book is provided elsewhere, on a short post (Feb. 2008). KNN regression is a non-parametric method that, in an intuitive manner, approximates the association between independent variables and the continuous outcome by averaging the observations in the same neighbourhood. 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) 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}\\ \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$, (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 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. 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 . More generally, The logistic regression model is overspecified. \mbox{additive model} &&\\ In terms of selecting the variables to model a particular response, four things can happen: A regression model is underspecified if it is missing one or more important predictor variables. $$\frac{L(p_0)}{L(\hat{p})}$$ gives us a sense of whether the null value or the observed value produces a higher likelihood. Suppose also that you know which topics each of your classmates is familiar with. &=& p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)}\\ p-value &=& P(\chi^2_1 \geq 190.16) = 0 P(X=1 | p = 0.75) &=& 0.047 \\ e^{\beta_1} &=& \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ H_1:&& p \ne 0.25\\ &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ The previous model specifies that the OR is constant for any value of $$X$$ which is not true about RR. P(X=1 | p = 0.9) &=& 0.0036 \\ 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. The authors cover t-tests, ANOVA and regression models, but also the advanced methods of generalised linear models and classification and regression … $\begin{eqnarray*} [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). ] 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. G: random guessing. \end{cases}\\ Agresti, A. 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. This dataset includes data taken from cancer.gov about deaths due to cancer in the United States. \end{eqnarray*}$, $\begin{eqnarray*} p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087\\ The results of HERS are surprising in light of previous observational studies, which found lower rates of CHD in women who take postmenopausal estrogen. \mathrm{logit}(\hat{p}) = 22.708 - 10.662 \cdot \ln(\mbox{ area }+1). H: is worse than random guessing. \end{eqnarray*}$. 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. z = \frac{b_1 - \beta_1}{SE(b_1)} 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)\\ 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. 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. That is, is the model able to discriminate between successes and failures. Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt But really, usually likelihood ratio tests are more interesting. In general, the method of least squares is applied to obtain the equation of the regression line. 0 & \mbox{ survived} Select the models based on the criteria we learned, as well as the number and nature of the predictors. If it guesses 90% of the positives correctly, it will also guess 90% of the negatives to be positive. What about the RR (relative risk) or difference in risks? G &=& 3594.8 - 3585.7= 9.1\\ H_1: && \beta_1 \ne 0\\ We locate the best variable, and regress the response variable on it. 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. $\begin{eqnarray*} \end{eqnarray*}$, When each person is at risk for a different covariate (i.e., explanatory variable), they each end up with a different probability of success. The majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. \mbox{sensitivity} &=& TPR = 300/308 = 0.974\\ The datasets below will be used throughout this course. The second type is MetaLasso, and our proposed method is as the third type. \end{eqnarray*}\], $\begin{eqnarray*} 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. Note 3: We can estimate any of the OR (of dying for smoke vs not smoke) from the given coefficients: $$\chi^2$$: The Likelihood ratio test also tests whether the response is explained by the explanatory variable. Recall that logistic regression can be used to predict the outcome of a binary event (your response variable). the negative-binomial regression model in DESeq2 (Love and others, 2014) and overdispersed Poisson model in edgeR (Robinson and others, 2010). 1 & \text{for always} \\ \mbox{interaction model} &&\\ \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 &=& 525.39 - 335.23 = 190.16\\ Evaluate the selected models for violation of the model conditions. \end{eqnarray*}$, $\begin{eqnarray*} This is bad. Regardless, we can see that by tuning the functional relationship of the S curve, we can get a good fit to the data. 0 & \text{otherwise} \\ In the survey, 2484 people were classified according to their proneness to snoring (never, occasionally, often, always) and whether or not they had the heart disease. && \\ Both techniques suggest choosing a model with the smallest AIC and BIC value; both adjust for the number of parameters in the model and are more likely to select models with fewer variables than the drop-in-deviance test. 1998). That is, a linear model as a function of the expected value of the response variable. There might be a few equally satisfactory models. Before we do that, we can define two criteria used for suggesting an optimal model. \ln L(p) &=& \ln \Bigg(p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)} \Bigg)\\ \end{eqnarray*}$, Our new model becomes: \end{eqnarray*}\], $\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} Some intuition of both calculus and Linear Algebra will make your journey easier. &=& \mbox{null (restricted) deviance - residual (full model) deviance}\\ p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 A good chunk of the cases were older, and the rate of smoking was substantially lower in the oldest group. The functional form relating x and the probability of success looks like it could be an S shape. 1 & \text{for occasionally} \\ e^{0} &=& 1\\ The 6th Seattle Symposium in Biostatistics will highlight Precision Health in the Age of Data Science, offering four thematic sessions, each with a keynote lecture, a selection of diverse talks, and a discussion panel, as well as optional preparatory short courses. Treating these topics together takes advantage of all they have in common. Note that tidy contains the same number of rows as the number of coefficients. G &=& 2 \cdot \ln(L(MLE)) - 2 \cdot \ln(L(null))\\ where $$\nu$$ represents the difference in the number of parameters needed to estimate in the full model versus the null model. The majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. \[\begin{eqnarray*} \mbox{young} & \mbox{18-44 years old}\\ However, the logit link (logistic regression) is only one of a variety of models that we can use. &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ If the observation corresponding to a survivor has a lower probability of success than the observation corresponding to a death, we call the pair discordant. If there are too many, we might just look at the correlation matrix. 1. “Snoring as a Risk Factor for Disease: An Epidemiological Survey” 291: 630–32. &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} 0 &=& (1-p) \sum_i y_i + p (n-\sum_i y_i) \\ &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ We start with the empty model, and add the best predictor, assuming the p-value associated with it is smaller than, Now, we find the best of the remaining variables, and add it if the p-value is smaller than. The Statistical Sleuth. \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} The training set, with at least 15-20 error degrees of freedom, is used to estimate the model. Recently, methods developed for RNA-seq data have been adapted to microbiome studies, e.g. G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} There’s not a data analyst out there who hasn’t made the mistake of skipping this step and later regretting it when a data point was found in error, thereby nullifying hours of work. For now, we will try to predict whether the individuals had a medical condition, medcond (defined as a pre-existing and self-reported medical condition). \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$ \end{eqnarray*}\]. x_1 &=& \begin{cases} 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. \beta_{1s} &=& \beta_1 + \beta_3 I can’t possibly over-emphasize the data exploration step. The method was based on multitask regression model enforced with sparse group For now, we will try to predict whether the individuals had a medical condition, medcond. 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. G &=& 3597.3 - 3594.8 =2.5\\ The examples, analyzed using Stata, are drawn from the biomedical context but generalize to other areas of application. 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). By using Kaggle, you agree to our use of cookies. &=& \sum_i y_i \ln(p) + (n- \sum_i y_i) \ln (1-p)\\ \end{eqnarray*}\]. \mbox{simple model} &&\\ Recall: x &=& \mbox{log area burned} These new methods can be used to perform prediction, estimation, and inference in complex big-data settings. &=& \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}}}\\ 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. \mbox{old} & \mbox{65+ years old}\\ Over 10 million scientific documents at your fingertips. p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \end{eqnarray*}\] \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ We will focus here only on model assessment. Co-organized by the Department of Biostatistics at the Harvard T.H. \end{eqnarray*}\] Y &\sim& \mbox{Bernoulli}(p)\\ \mbox{additive model} &&\\ \end{eqnarray*}\]. 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. p(k) &=& 1-(1-\lambda)^k\\ p(-\beta_0 / \beta_1) &=& p(x) = 0.5 $\begin{eqnarray*} \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 && \\ \mbox{middle} & \mbox{45-64 years old}\\ where $$\hat{\beta}_0$$ is fit from a model without any explanatory variable, $$x$$. \mbox{interaction model} &&\\ Applied Logistic Regression is an ideal choice." \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) \end{eqnarray*}$, $\begin{eqnarray*} 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. There would probably be a different slope for each class year in order to model the two variables most effectively. \hat{p}(2.5) &=& 0.01894664\\ \[\begin{eqnarray*} advantage of integrating multiple diverse datasets over analyzing them individually. \end{eqnarray*}$, $\begin{eqnarray*} The participants are postmenopausal women with a uterus and with CHD. Another strategy for model building. 2 Several methods that remove or adjust batch variation have been developed. Figure taken from (Ramsey and Schafer 2012). &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ GLM: g(E[Y | X]) = \beta_0 + \beta_1 X \end{eqnarray*}$. \ln[ - \ln (1-p(k))] &=& \beta_0 + \beta_1 x\\ Decide on the type of model that is needed in order to achieve the goals of the study. We do have good reasons for how we defined it, but that doesn’t mean there aren’t other good ways to model the relationship.). 1 & \mbox{ smoke}\\ &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ gives the odds of success. The patients were grouped according to the area of third-degree burns on the body (measured in square cm). \end{eqnarray*}\], $\begin{eqnarray*} \hat{p} &=& \frac{ \sum_i y_i}{n} \mathrm{logit}(p(x_1, x_2) ) &=& \beta_0 + \beta_1 x_1 + \beta_2 x_2\\ “Randomized Trial of Estrogen Plus Progestin for Secondary Prevention of Coronary Heart Disease in Postmenopausal Women.” Journal of the American Medical Association 280: 605–13. \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$. \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) http://statmaster.sdu.dk/courses/st111. $\begin{eqnarray*} E[\mbox{grade seniors}| \mbox{hours studied}] &=& \beta_{0s} + \beta_{1s} \mbox{hrs}\\ http://www.r2d3.us/visual-intro-to-machine-learning-part-1/. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ To maximize the likelihood, we use the natural log of the likelihood (because we know we’ll get the same answer): 5, 6 Undetected batch effects can have major impact on subsequent conclusions in both unsupervised and supervised analysis. \end{eqnarray*}$, $\begin{eqnarray*} \[\begin{eqnarray*} If we are testing only one parameter value. p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ Bayesian and Frequentist Regression Methods Website. \[\begin{eqnarray*} A brief introduction to regression analysis of complex surveys and notes for further reading are provided. Covers all of the nuts and bolts of biostatistics in a user-friendly style that motivates readers. You begin by trying to answer the questions from previous papers and comparing your answers with the model answers provided. In the last model, we might want to remove all the age information. Menard, S. 1995. &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ && \\ \end{eqnarray*}$ tau-a: Kendall’s tau-a is the number of concordant pairs minus the number of discordant pairs divided by the total number of pairs of people (including pairs who both survived or both died). \end{eqnarray*}\], $\begin{eqnarray*} On a univariate basis, check for outliers, gross data errors, and missing values. \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ Some intuition of both calculus and Linear Algebra will make your journey easier. \mathrm{logit}(p(x)) &=& \beta_0 + \beta_1 x\\ \end{eqnarray*}$ \mathrm{logit}(\star) = \ln \bigg( \frac{\star}{1-\star} \bigg) \ \ \ \ 0 < \star < 1 \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 Ours is called the logit. (The logistic model is just one model, there isn’t anything magical about it. What does that even mean? &=& \ln \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ 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. 2nd ed. \frac{p(x)}{1-p(x)} = e^{\beta_0 + \beta_1 x} P(X=1 | p) &=& {4 \choose 1} p^1 (1-p)^{4-1}\\