# regression methods in biostatistics datasets

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. \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ \end{eqnarray*}\], $\begin{eqnarray*} That is, the odds of survival for a patient with log(area+1)= 1.90 is 2.9 times higher than the odds of survival for a patient with log(area+1)= 2.0.). The problem with this strategy is that it may be that the 75 subjects Bruno knows are already included in the 85 that Sage knows, and therefore, Bruno does not provide any knowledge beyond that of Sage. We applied three types of methods to these two datasets. This course provides basic knowledge of logistic regression and analysis of survival data. \end{eqnarray*}$ It seems that a transformation of the data is in place. p-value &=& P(\chi^2_1 \geq 190.16) = 0 We can output the deviance ( = K - 2 * log-likelihood) for both the full (maximum likelihood!) p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ A short summary of the book is provided elsewhere, on a short post (Feb. 2008). \end{eqnarray*}\]. BIC: Bayesian Information Criteria = $$-2 \ln$$ likelihood $$+p \ln(n)$$. $\begin{eqnarray*} The logistic regression model is correct! \mathrm{logit}(p(x_1, x_2) ) &=& \beta_0 + \beta_1 x_1 + \beta_2 x_2\\ We will study Linear Regression, Polynomial Regression, Normal equation, gradient descent and step by step python implementation. The explanatory variable of interest was the length of the bird. Applied Logistic Regression is an ideal choice." \[\begin{eqnarray*} Ramsey, F., and D. Schafer. \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) \mbox{old} & \mbox{65+ years old}\\ \end{cases} The datasets below will be used throughout this course. \end{eqnarray*}$ While a first course in statistics is assumed, a chapter reviewing basic statistical methods is included. Supplemented with numerous graphs, charts, and tables as well as a Web site for larger data sets and exercises, Biostatistical Methods: The Assessment of Relative Risks is an excellent guide for graduate-level students in biostatistics and an invaluable reference for biostatisticians, applied statisticians, and epidemiologists. \end{eqnarray*}\], $\begin{eqnarray*} A Receiver Operating Characteristic (ROC) Curve is a graphical representation of the relationship between. 0 &=& (1-p) \sum_i y_i + p (n-\sum_i y_i) \\ Being underspecified is the worst case scenario because the model ends up being biased and predictions are wrong for virtually every observation. This method of estimating the parameters of a regression line is known as the method of least squares. \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) A: Let’s say we use prob=0.25 as a cutoff: \[\begin{eqnarray*} Bayesian and Frequentist Regression Methods (Springer Series in Statistics) - Kindle edition by Wakefield, Jon. Decide on the type of model that is needed in order to achieve the goals of the study. Note 3 gives the probability of failure. Continue removing variables until all variables are significant at the chosen. &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ The validation set is used for cross-validation of the fitted model. 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}\\ x_1 &=& \begin{cases} \end{eqnarray*}$, $\begin{eqnarray*} H_0: && \beta_1 =0\\ Note that the opposite classifier to (H) might be quite good! \[\begin{eqnarray*} where $$\nu$$ is the number of extra parameters we estimate using the unconstrained likelihood (as compared to the constrained null likelihood). Wand. \[\begin{eqnarray*} 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. To maximize the likelihood, we use the natural log of the likelihood (because we know we’ll get the same answer): x &=& - \beta_0 / \beta_1\\ 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. \end{eqnarray*}$, $\begin{eqnarray*} There might be a few equally satisfactory models. \beta_0 + \beta_1 x &=& 0\\ Recall how we estimated the coefficients for linear regression. &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ \end{eqnarray*}$, $\begin{eqnarray*} \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ Ours is called the logit. \beta_{1s} &=& \beta_1 + \beta_3 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. \end{cases} \mbox{simple model} &&\\ \mbox{specificity} &=& 61 / 127 = 0.480, \mbox{1 - specificity} = FPR = 0.520\\ The odds ratio $$\hat{OR}_{1.90, 2.00}$$ is given by p(k) &=& 1-(1-\lambda)^k\\ Not logged in A brief introduction to regression analysis of complex surveys and notes for further reading are provided. Linear Regression Datasets for Machine Learning. x_1 &=& \begin{cases} where $$y_1, y_2, \ldots, y_n$$ represents a particular observed series of 0 or 1 outcomes and $$p$$ is a probability $$0 \leq p \leq 1$$. &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ The participants are postmenopausal women with a uterus and with CHD. It gives you a sense of whether or not you’ve overfit the model in the building process.) H_0: && \beta_1 =0\\ &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ \mbox{specificity} &=& 120/127 = 0.945, \mbox{1 - specificity} = FPR = 0.055\\ e^{\beta_1} &=& \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ \end{eqnarray*}$, $\begin{eqnarray*} The table below shows the result of the univariate analysis for some of the variables in the dataset. Model building is definitely an art." \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} (SBH). P(X=1 | p = 0.9) &=& 0.0036 \\ How is it interpreted? It turns out that we’ve also maximized the normal likelihood. &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 &=& p^{y_1}(1-p)^{1-y_1} p^{y_2}(1-p)^{1-y_2} \cdots p^{y_n}(1-p)^{1-y_n}\\ 1 - p(x) = \frac{1}{1+e^{\beta_0 + \beta_1 x}} “Snoring as a Risk Factor for Disease: An Epidemiological Survey” 291: 630–32. That is because age and smoking status are so highly associated (think of the coin example). Consider the following data set collected from church offering plates in 62 consecutive Sundays. \end{eqnarray*}$, $\begin{eqnarray*} L(\hat{\underline{p}}) > L(p_0) An Introduction to Categorical Data Analysis. Statistics for Biology and Health &=& \sum_i y_i \ln(p) + (n- \sum_i y_i) \ln (1-p)\\ where $$\nu$$ represents the difference in the number of parameters needed to estimate in the full model versus the null model. The logistic regression model is a generalizedlinear model. \end{eqnarray*}$, $\begin{eqnarray*} 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. Regardless, we can see that by tuning the functional relationship of the S curve, we can get a good fit to the data. \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ \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)\\ advantage of integrating multiple diverse datasets over analyzing them individually. \end{eqnarray*}$, $\begin{eqnarray*} Start with the full model including every term (and possibly every interaction, etc.). Recall, when comparing two nested models, the differences in the deviances can be modeled by a $$\chi^2_\nu$$ variable where $$\nu = \Delta p$$. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ \ln (p(x)) = \beta_0 + \beta_1 x Where $$p(x)$$ is the probability of success (here surviving a burn). 3rd ed. Also problematic is that the model becomes unnecessarily complicated and harder to interpret. \mbox{middle} & \mbox{45-64 years old}\\ 5, 6 Undetected batch effects can have major impact on subsequent conclusions in both unsupervised and supervised analysis. 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. Before we do that, we can define two criteria used for suggesting an optimal model. 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). Consider false positive rate, false negative rate, outliers, parsimony, relevance, and ease of measurement of predictors. $$e^{\beta_1}$$ is the odds ratio for dying associated with a one unit increase in x. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. Bayesian and Frequentist Regression Methods Website. In general, the method of least squares is applied to obtain the equation of the regression line. \end{eqnarray*}$, $\begin{eqnarray*} For logistic regression, we use the logit link function: We see above that the logistic model imposes a constant OR for any value of $$X$$ (and not a constant RR). Y &\sim& \mbox{Bernoulli}(p)\\ Tied pairs occur when the observed survivor has the same estimated probability as the observed death. Over 10 million scientific documents at your fingertips. p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ and reduced (null) models. Select the models based on the criteria we learned, as well as the number and nature of the predictors. Let’s say this is Sage who knows 85 topics. \end{eqnarray*}$ &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 B: Let’s say we use prob=0.7 as a cutoff: $\begin{eqnarray*} Example 5.4 Suppose that you have to take an exam that covers 100 different topics, and you do not know any of them. \[\begin{eqnarray*} &=& -2 \Bigg[ \ln \bigg( (0.25)^{y} (0.75)^{n-y} \bigg) - \ln \Bigg( \bigg( \frac{y}{n} \bigg)^{y} \bigg( \frac{(n-y)}{n} \bigg)^{n-y} \Bigg) \Bigg]\\ p(x) &=& \beta_0 + \beta_1 x However, the scatterplot of the proportions of patients surviving a third-degree burn against the explanatory variable shows a distinct curved relationship between the two variables, rather than a linear one. This new book provides a unified, in-depth, readable introduction to the multipredictor regression methods most widely used in biostatistics: linear models for continuous outcomes, logistic models for binary outcomes, the Cox model for right-censored survival times, repeated-measures models for longitudinal and hierarchical outcomes, and generalized linear models for counts and other outcomes. Note 4 Every type of generalized linear model has a link function. They are: Decide which explanatory variables and response variable on which to collect the data. \end{eqnarray*}$, $\begin{eqnarray*} 2012. \end{eqnarray*}$ 0 & \text{otherwise} \\ The big model (with all of the interaction terms) has a deviance of 3585.7; the additive model has a deviance of 3594.8. \end{cases} This dataset includes data taken from cancer.gov about deaths due to cancer in the United States. \end{eqnarray*}\], $\begin{eqnarray*} Multivariable logistic regression. We can show that if $$H_0$$ is true, 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^*}}\\ How do we model? (The fourth step is very good modeling practice. x_2 &=& \begin{cases} Graduate Prerequisites: The biostatistics and epidemiology MPH core course requirements and BS723 or BS852. &=& -2 [ \ln(0.0054) - \ln(0.0697) ] = 5.11\\ Maximum likelihood estimates are functions of sample data that are derived by finding the value of $$p$$ that maximizes the likelihood functions. Using the burn data, convince yourself that the RR isn’t constant. &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ Evaluate the selected models for violation of the model conditions. Introductory course in the analysis of Gaussian and categorical data. G &=& 525.39 - 335.23 = 190.16\\ http://www.r2d3.us/visual-intro-to-machine-learning-part-1/. \end{eqnarray*}$. It is quite common to have binary outcomes (response variable) in the medical literature. \beta_{1f} &=& \beta_1\\ Not affiliated Hulley, S., D. Grady, T. Bush, C. Furberg, D. Herrington, B. Riggs, and E. Vittinghoff. John Wiley; Sons, New York. \end{eqnarray*}\], $\begin{eqnarray*} We will use the variables age, weight, diabetes and drinkany. What does it mean that the interaction terms are not significant in the last model? http://statmaster.sdu.dk/courses/st111. If classifier randomly guess, it should get half the positives correct and half the negatives correct. Or, we can think about it as a set of independent binary responses, $$Y_1, Y_2, \ldots Y_n$$. \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$. They also show that these regression methods deal with confounding, mediation, and interaction of causal effects in essentially the same way. Part of Springer Nature. This service is more advanced with JavaScript available, Part of the Using the additive model above: -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu The second type is MetaLasso, and our proposed method is as the third type. Sage. In many situations, this will help us from stopping at a less than desirable model. && \\ GLM: g(E[Y | X]) = \beta_0 + \beta_1 X \mbox{& a loglikelihood of}: &&\\ P( \chi^2_1 \geq 5.11) &=& 0.0238 \end{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) © 2020 Springer Nature Switzerland AG. \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 L(\hat{\underline{p}}) > L(p_0) We are going to discuss how to add (or subtract) variables from a model. 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. $\begin{eqnarray*} A large cross-validation AUC on the validation data is indicative of a good predictive model (for your population of interest). If it guesses 90% of the positives correctly, it will also guess 90% of the negatives to be positive. If there are too many, we might just look at the correlation matrix. The training set, with at least 15-20 error degrees of freedom, is used to estimate the model. The course will cover extensions of these methods to correlated data using generalized estimating equations. \frac{p(x)}{1-p(x)} = e^{\beta_0 + \beta_1 x} \end{eqnarray*}$ RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ \end{eqnarray*}\] \end{cases}\\ RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ We can estimate the SE (Wald estimates via Fisher Information). Datasets Most of the datasets on this page are in the S dumpdata and R compressed save() file formats. \end{eqnarray*}\], $\begin{eqnarray*} \[\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}\\ \mbox{interaction model} &&\\ e^{0} &=& 1\\ 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. Contains notes on computations at the end of most chapters, covering the use of Excel, SAS, and others. 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. p(x) &=& 1 - \exp [ -\exp(\beta_0 + \beta_1 x) ] 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. We use cookies on Kaggle to deliver our services, analyze web traffic, and improve your experience on the site. The method was based on multitask regression model enforced with sparse group \[\begin{eqnarray*} (see log-linear model below, 5.1.2.1 ). \[\begin{eqnarray*} The second half introduces bivariate and multivariate methods, emphasizing contingency table analysis, regression, and analysis of variance. For now, we will try to predict whether the individuals had a medical condition, medcond. \end{eqnarray*}$, $\begin{eqnarray*} Consider looking at all the pairs of successes and failures. (Agresti 1996) states that the likelihood-ratio test is more reliable for small sample sizes than the Wald test. 1 & \mbox{ smoke}\\ Dunn. To build a model (model selection). 0 & \text{otherwise} \\ \mbox{sensitivity} &=& TPR = 265/308 = 0.860\\ \[\begin{eqnarray*} The logistic regression model is overspecified. 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. However, within each group, the cases were more likely to smoke than the controls. Generally, extraneous variables are not so problematic because they produce models with unbiased coefficient estimators, unbiased predictions, and unbiased variance estimates. 1995. 1985. \[\begin{eqnarray*} One idea is to start with an empty model and adding the best available variable at each iteration, checking for needs for transformations. P(X=1 | p = 0.5) &=& 0.25\\ G &=& 3597.3 - 3594.8 =2.5\\ \end{eqnarray*}$ && \\ Note that the x-axis is some continuous variable x while the y-axis is the probability of success at that value of x. biostat/vgsm/data/hersdata.txt, and it is described in Regression Methods in Biostatistics, page 30; variable descriptions are also given on the book website http://www.epibiostat.ucsf.edu/biostat/ vgsm/data/hersdata.codebook.txt. \end{eqnarray*}\], $\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} x_2 &=& \begin{cases} \end{eqnarray*}$ That is, a linear model as a function of the expected value of the response variable. &=& \bigg( \frac{1}{2 \pi \sigma^2} \bigg)^{n/2} e^{\sum_i (y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ \hat{RR}_{1, 2} &=& 1.250567\\ Recall that logistic regression can be used to predict the outcome of a binary event (your response variable). \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ Though it is important to realize that we cannot find estimates in closed form. P(X=1 | p = 0.9) &=& 0.0036 \\ \mbox{interaction model} &&\\ The methods introduced include robust estimation, testing, model selection, model check and diagnostics. (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 … p_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} Cengage Learning. \end{eqnarray*}\]. Is a different picture provided by considering odds? Suppose that we build a classifier (logistic regression model) on a given data set. What does that even mean? 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. 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. \end{cases} 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. \end{eqnarray*}\], $\begin{eqnarray*} For control purposes - that is, the model will be used to control a response variable by manipulating the values of the predictor variables. Note that tidy contains the same number of rows as the number of coefficients. Fan, J., N.E. 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 & \text{for always} \\ (Think about Simpson’s Paradox and the need for interaction.). 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. $$\beta_0$$ now determines the location (median survival). 0 & \mbox{ don't smoke}\\ augment contains the same number of rows as number of observations. Another strategy for model building. Therefore, if its possible, a scatter plot matrix would be best. 1 & \text{for often} \\ \hat{p} &=& \frac{ \sum_i y_i}{n} 2nd ed. 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). \mathrm{logit}(p(x)) &=& \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. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. \end{eqnarray*}$, Our new model becomes: H_0:&& p=0.25\\ In fact, usually, we use them to test whether the coefficients are zero: $\begin{eqnarray*} -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu \mbox{specificity} &=& 92/127 = 0.724, \mbox{1 - specificity} = FPR = 0.276\\ \[\begin{eqnarray*}GLM: g(E[Y | X]) = \beta_0 + \beta_1 X\end{eqnarray*}$where $$g(\cdot)$$is the … 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). \ln[ - \ln (1-p(k))] &=& \ln[-\ln(1-\lambda)] + \ln(k)\\ 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.