# regression methods in biostatistics datasets

The first half of the course introduces descriptive statistics and the inferential statistical methods of confidence intervals and significance tests. gives the probability of failure. p_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 The previous model specifies that the OR is constant for any value of $$X$$ which is not true about RR. $\begin{eqnarray*} where $$\hat{\beta}_0$$ is fit from a model without any explanatory variable, $$x$$. 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^*}}\\ The participants are postmenopausal women with a uterus and with CHD. These methods, however, are not optimized for microbiome datasets. We should also look at interactions which we might suspect. In general, the method of least squares is applied to obtain the equation of the regression line. The first type of method applied logistic regression model with the four penalties to the merged data directly. It won’t be constant for a given $$X$$, so it must be calculated as a function of $$X$$. Recall: However, we may miss out of variables that are good predictors but aren’t linearly related. What we see is that the vast majority of the controls were young, and they had a high rate of smoking. P( \chi^2_1 \geq 5.11) &=& 0.0238 Figure taken from (Ramsey and Schafer 2012). We can now model binary response variables. \end{eqnarray*}$, Our new model becomes: Recall: But we’d have to do some work to figure out what the form of that S looks like. Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt \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)}}}\\ $\begin{eqnarray*} Lesson of the story: be very very very careful interpreting coefficients when you have multiple explanatory variables. 2012. \end{eqnarray*}$, $\begin{eqnarray*} p(k) &=& 1-(1-\lambda)^k\\ What about the RR (relative risk) or difference in risks? The functional form relating x and the probability of success looks like it could be an S shape. \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) Regression modeling of categorical or time-to-event outcomes with continuous and categorical predictors is covered. http://www.r2d3.us/visual-intro-to-machine-learning-part-1/. \end{eqnarray*}$, $\begin{eqnarray*} 2nd ed. \end{eqnarray*}$, $\begin{eqnarray*} Applications Required; Filetype Application.mtw: Minitab / Minitab Express (recommended).xls, .xlsx: Microsoft Excel / Alternatives.txt We use cookies on Kaggle to deliver our services, analyze web traffic, and improve your experience on the site. How do you choose the $$\alpha$$ values? \mbox{test stat} &=& \chi^2\\ For simplicity, consider only first year students and seniors. 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. We locate the best variable, and regress the response variable on it. \[\begin{eqnarray*} &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ A brief introduction to regression analysis of complex surveys and notes for further reading are provided. We can, however, measure whether or not the estimated model is consistent with the data. P(X=1 | p = 0.75) &=& 0.047 \\ Would you guess $$p=0.49$$?? e^{\beta_1} &=& \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ Cengage Learning. \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ Though it is important to realize that we cannot find estimates in closed form. \end{eqnarray*}$, $\begin{eqnarray*} \mathrm{logit}(p(x+1)) &=& \beta_0 + \beta_1 (x+1)\\ What does that even mean? The pairs would be concordant if the first individual survived and the second didn’t. \[\begin{eqnarray*}GLM: g(E[Y | X]) = \beta_0 + \beta_1 X\end{eqnarray*}$where $$g(\cdot)$$is the … Recall how we estimated the coefficients for linear regression. 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. The logistic regression model is a generalizedlinear model. Before reading the notes here, look through the following visualization. -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) &=& -2 [ \ln (L(p_0)) - \ln(L(\hat{p}))]\\ That is, the variables contain the same information as other variables (i.e., are correlated!). 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). &=& -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]\\ 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. \end{eqnarray*}\], $\begin{eqnarray*} As done previously, we can add and remove variables based on the deviance. &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ Let’s say $$X \sim Bin(p, n=4).$$ We have 4 trials and $$X=1$$. The examples, analyzed using Stata, are drawn from the biomedical context but generalize to other areas of application. Download it once and read it on your Kindle device, PC, phones or tablets. 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. Use stepwise regression, which of course only yields one model unless different alpha-to-remove and alpha-to-enter values are specified. \mathrm{logit}(\star) = \ln \bigg( \frac{\star}{1-\star} \bigg) \ \ \ \ 0 < \star < 1 \[\begin{eqnarray*} If there are too many, we might just look at the correlation matrix. \ln L(p) &=& \ln \Bigg(p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)} \Bigg)\\ \end{eqnarray*}$. Example 4.3 Consider a simple linear regression model on number of hours studied and exam grade. \beta_{1s} &=& \beta_1 + \beta_3 http://statmaster.sdu.dk/courses/st111. BIOST 570 Advanced Regression Methods for Independent Data (3) Covers linear models, generalized linear and non-linear regression, and models. \end{eqnarray*}\] 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. \hat{RR}_{1.5, 2.5} &=& 52.71587\\ Dunn. p-value &=& P(\chi^2_1 \geq 190.16) = 0 $\begin{eqnarray*} \[\begin{eqnarray*} p(0) = \frac{e^{\beta_0}}{1+e^{\beta_0}} [$$\beta_1$$ is the change in log-odds associated with a one unit increase in x. 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. Since each observed response is independent and follows the Bernoulli distribution, the probability of a particular outcome can be found as: In the last model, we might want to remove all the age information. \mbox{additive model} &&\\ That is, for one level of a variable, the relationship of the main predictor on the response is different. p_0 &=& \frac{e^{\hat{\beta}_0}}{1 + e^{\hat{\beta}_0}} The logistic regression model is overspecified. \mbox{old} & \mbox{65+ years old}\\ Where $$p(x)$$ is the probability of success (here surviving a burn). For control purposes - that is, the model will be used to control a response variable by manipulating the values of the predictor variables. data described in Breslow and Day (1980) from a matched case control study. BIC: Bayesian Information Criteria = $$-2 \ln$$ likelihood $$+p \ln(n)$$. \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x \end{eqnarray*}$, Let’s say the log odds of survival for given observed (log) burn areas $$x$$ and $$x+1$$ are: \beta_0 + \beta_1 x &=& 0\\ Now, if the upcoming exam completely consists of past questions, you are certain to do very well. where $$\nu$$ represents the difference in the number of parameters needed to estimate in the full model versus the null model. \frac{p(x)}{1-p(x)} = e^{\beta_0 + \beta_1 x} &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ \end{eqnarray*}\], $\begin{eqnarray*} Robust Methods in Biostatistics proposes robust alternatives to common methods used in statistics in general and in biostatistics in particular and illustrates their use on many biomedical datasets. Also noted is whether there was enough change to buy a candy bar for 1.25. 1998. advantage of integrating multiple diverse datasets over analyzing them individually. P(Y_1=y_1, Y_2=y_2, \ldots, Y_n=y_n) &=& P(Y_1=y_1) P(Y_2 = y_2) \cdots P(Y_n = y_n)\\ \end{eqnarray*}$, $\begin{eqnarray*} P(Y_1=y_1, Y_2=y_2, \ldots, Y_n=y_n) &=& P(Y_1=y_1) P(Y_2 = y_2) \cdots P(Y_n = y_n)\\ This new book provides a unified, in-depth, readable introduction to the multipredictor regression methods most widely used in biostatistics: linear models for continuous outcomes, logistic models for binary outcomes, the Cox model for right-censored survival times, repeated-measures models for longitudinal and hierarchical outcomes, and generalized linear models for counts and other outcomes. G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} Being underspecified is the worst case scenario because the model ends up being biased and predictions are wrong for virtually every observation. P(X=1 | p = 0.9) &=& 0.0036 \\ \end{eqnarray*}$, $\begin{eqnarray*} P(Y=y) &=& p^y(1-p)^{1-y} As we’ve seen, correlated variables cause trouble because they inflate the variance of the coefficient estimates. Advanced Methods in Biostatistics IV - Regression Modeling Advanced Methods in Biostatistics IV covers topics in modern multivariate regression from estimation theoretic, likelihood-based, and Bayesian points of view. Introductory course in the analysis of Gaussian and categorical data. X_1 = \begin{cases} There would probably be a different slope for each class year in order to model the two variables most effectively. book series \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} When a model is overspecified, there are one or more redundant variables. Tied pairs occur when the observed survivor has the same estimated probability as the observed death. While a first course in statistics is assumed, a chapter reviewing basic statistical methods is included. 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 authors cover t-tests, ANOVA and regression models, but also the advanced methods of generalised linear models and classification and regression … P(X=1 | p = 0.5) &=& 0.25\\ E[\mbox{grade first years}| \mbox{hours studied}] &=& \beta_{0f} + \beta_{1f} \mbox{hrs}\\ The Statistical Sleuth. That is, is the model able to discriminate between successes and failures. 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) p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ \end{cases} \end{eqnarray*}$, $\begin{eqnarray*} \[\begin{eqnarray*} Notice that the directionality of the low coins changes when it is included in the model that already contains the number of coins total. A: Let’s say we use prob=0.25 as a cutoff: \[\begin{eqnarray*} Introductory course in the analysis of Gaussian and categorical data. \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ It gives you a sense of whether or not you’ve overfit the model in the building process.) \[\begin{eqnarray*} Deep dive into Regression Analysis and how we can use this to infer mindboggling insights using Chicago COVID dataset. The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. This course provides basic knowledge of logistic regression and analysis of survival data. \end{eqnarray*}$ On a univariate basis, check for outliers, gross data errors, and missing values. Deep dive into Regression Analysis and how we can use this to infer mindboggling insights using Chicago COVID dataset. Taken from https://onlinecourses.science.psu.edu/stat501/node/332. \end{cases}\\ Unfortunately, you get carried away and spend all your time on memorizing the model answers to all past questions. Fan, J., N.E. \end{cases} \end{eqnarray*}\]. \mbox{sensitivity} &=& TPR = 144/308 = 0.467\\ 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). p(x) &=& 1 - \exp [ -\exp(\beta_0 + \beta_1 x) ] This method follows in the same way as Forward Regression, but as each new variable enters the model, we check to see if any of the variables already in the model can now be removed. 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. \end{eqnarray*}\], $\begin{eqnarray*} \[\begin{eqnarray*} x &=& \mbox{log area burned} Note 3 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). When using our method, we set μ=1 and α=0.5 except LASSO penalty. 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. \end{eqnarray*}$, $\begin{eqnarray*} L(\underline{y} | b_0, b_1, \underline{x}) &=& \prod_i \frac{1}{\sqrt{2 \pi \sigma^2}} e^{(y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. \ln[ - \ln (1-p(k))] &=& \ln[-\ln(1-\lambda)] + \ln(k)\\ \end{eqnarray*}$ Introductory course in the analysis of Gaussian and categorical data. x_2 &=& \begin{cases} \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. &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ The majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. Example 5.3 Consider the example on smoking and 20-year mortality (case) from section 3.4 of Regression Methods in Biostatistics, pg 52-53. $\begin{eqnarray*} The hormone replacement regimen also increased the risk of clots in the veins (deep vein thrombosis) and lungs (pulmonary embolism). In particular, methods are illustrated using a variety of data sets. In particular, methods are illustrated using a variety of data sets. Let’s say this is Sage who knows 85 topics. \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ [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). ] Part of Springer Nature. The validation set is used for cross-validation of the fitted model. 1 & \mbox{ died}\\ \end{cases}\\ G &=& 525.39 - 335.23 = 190.16\\ Regardless, we can see that by tuning the functional relationship of the S curve, we can get a good fit to the data. The estimates have an approximately normal sampling distribution for large sample sizes because they are maximum likelihood estimates. \end{eqnarray*}$, $\begin{eqnarray*} Decide on the type of model that is needed in order to achieve the goals of the study. OR &=& \mbox{odds dying if } (x_1, x_2) / \mbox{odds dying if } (x_1^*, x_2^*) = \frac{e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2}}{e^{\beta_0 + \beta_1 x_1^* + \beta_2 x_2^*}}\\ RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ \end{eqnarray*}$ Recall, when comparing two nested models, the differences in the deviances can be modeled by a $$\chi^2_\nu$$ variable where $$\nu = \Delta p$$. \end{eqnarray*}\] &=& \sum_i y_i \ln(p) + (n- \sum_i y_i) \ln (1-p)\\ p-value &=& P(\chi^2_1 \geq 2.5)= 1 - pchisq(2.5, 1) = 0.1138463 \ln \bigg( \frac{p(x)}{1-p(x)} \bigg) = \beta_0 + \beta_1 x However, the logit link (logistic regression) is only one of a variety of models that we can use. 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. G &\sim& \chi^2_{\nu} \ \ \ \mbox{when the null hypothesis is true} \end{eqnarray*}\] The course will cover extensions of these methods to correlated data using generalized estimating equations. An Introduction to Categorical Data Analysis. which gives a likelihood of: \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)\\ 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. H_0: && \beta_1 =0\\ \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ \mbox{old} & \mbox{65+ years old}\\ \end{eqnarray*}\]. 1. \end{eqnarray*}\]. These new methods can be used to perform prediction, estimation, and inference in complex big-data settings. 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 big model (with all of the interaction terms) has a deviance of 3585.7; the additive model has a deviance of 3594.8. &=& \mbox{deviance}_{reduced} - \mbox{deviance}_{full}\\ glance always has one row (containing overall model information). The logistic regression model is underspecified. 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. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ &=& 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*} The additive model has a deviance of 3594.8; the model without weight is 3597.3. This method of estimating the parameters of a regression line is known as the method of least squares. \[\begin{eqnarray*} More generally, x_2 &=& \begin{cases} \ln[ - \ln (1-p(k))] &=& \beta_0 + 1 \cdot \ln(k)\\ Not logged in That is because age and smoking status are so highly associated (think of the coin example). Generally, extraneous variables are not so problematic because they produce models with unbiased coefficient estimators, unbiased predictions, and unbiased variance estimates. \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$ The table below shows the result of the univariate analysis for some of the variables in the dataset. to log(area +1)= 2.00. ), things can get out of hand quickly. &=& \ln \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ Linear Regression Datasets for Machine Learning. \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) After adjusting for age, smoking is no longer significant. &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ To build a model (model selection). \hat{p}(1.5) &=& 0.9987889\\ Agresti, A. We continue with this process until there are no more variables that meet either requirements. Some intuition of both calculus and Linear Algebra will make your journey easier. This occurred despite the positive effect of treatment on lipoproteins: LDL (bad) cholesterol was reduced by 11 percent and HDL (good) cholesterol was increased by 10 percent. Once $$y_1, y_2, \ldots, y_n$$ have been observed, they are fixed values. Y &\sim& \mbox{Bernoulli}(p)\\ \end{eqnarray*}\] and specificity (no FP). It seems that a transformation of the data is in place. p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} 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 will use the variables age, weight, diabetes and drinkany. \end{eqnarray*}\], So, the LRT here is (see columns of null deviance and deviance): P(X=1 | p = 0.9) &=& 0.0036 \\ For logistic regression, we use the logit link function: \mbox{sensitivity} &=& TPR = 265/308 = 0.860\\ \beta_{1f} &=& \beta_1\\ p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} Contains notes on computations at the end of most chapters, covering the use of Excel, SAS, and others. 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. 1 & \mbox{ smoke}\\ &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ \mathrm{logit}(p) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 \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 \hat{p}(2.5) &=& 0.01894664\\ \hat{p(x)} &=& \frac{e^{22.708 - 10.662 x}}{1+e^{22.708 - 10.662 x}}\\ $\begin{eqnarray*} For predictive reasons - that is, the model will be used to predict the response variable from a chosen set of predictors. \end{eqnarray*}$. \mbox{young} & \mbox{18-44 years old}\\ (Think about Simpson’s Paradox and the need for interaction.). &=& \ln \bigg( \frac{p(x+1) / [1-p(x+1)]}{p(x) / [1-p(x)]} \bigg)\\ Just like in linear regression, our Y response is the only random component. ], leave one out cross validation (LOOCV) [LOOCV is a special case of, build the model using the remaining n-1 points, predict class membership for the observation which was removed, repeat by removing each observation one at a time (time consuming to keep building models), like LOOCV except that the algorithm is run. $\begin{eqnarray*} That is, a linear model as a function of the expected value of the response variable. \[\begin{eqnarray*} 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. Statistical tools for analyzing experiments involving genomic data. 0 & \text{otherwise} \\ The logistic regression model contains extraneous variables. Evaluate the selected models for violation of the model conditions. &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ P(X=1 | p = 0.05) &=& 0.171\\ We are going to discuss how to add (or subtract) variables from a model. 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. G &=& 3597.3 - 3594.8 =2.5\\ We will study Linear Regression, Polynomial Regression, Normal equation, gradient descent and step by step python implementation. 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). That is, the variables are important in predicting odds of survival. Suppose that we build a classifier (logistic regression model) on a given data set. 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). \end{eqnarray*}$, $\begin{eqnarray*} x_1 &=& \begin{cases} Some intuition of both calculus and Linear Algebra will make your journey easier. \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. 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). Consider the following data set collected from church offering plates in 62 consecutive Sundays. 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(\hat{\underline{p}}) > L(p_0) Note that the opposite classifier to (H) might be quite good! The results of HERS are surprising in light of previous observational studies, which found lower rates of CHD in women who take postmenopausal estrogen. \end{eqnarray*}\], $\begin{eqnarray*} Therefore, if its possible, a scatter plot matrix would be best. \hat{p}(2) &=& 0.7996326\\ \hat{OR}_{1.90, 2.00} = e^{-10.662} (1.90-2.00) = e^{1.0662} = 2.904 Hulley, S., D. Grady, T. Bush, C. Furberg, D. Herrington, B. Riggs, and E. Vittinghoff. 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. 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. Imagine you are preparing for your statistics exam. gives the $$\ln$$ odds of success . © 2020 Springer Nature Switzerland AG. &=& \frac{1+e^{b_0}e^{b_1 x}e^{b_1}}{e^{b_1}(1+e^{b_0}e^{b_1 x})}\\ We will focus here only on model assessment. \[\begin{eqnarray*} GLM: g(E[Y | X]) = \beta_0 + \beta_1 X && \\ 1995. &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ Maximum likelihood estimates are functions of sample data that are derived by finding the value of $$p$$ that maximizes the likelihood functions. However, (Menard 1995) warns that for large coefficients, standard error is inflated, lowering the Wald statistic (chi-square) value. G &=& 2 \cdot \ln(L(MLE)) - 2 \cdot \ln(L(null))\\ \end{eqnarray*}$ \end{eqnarray*}\], $\begin{eqnarray*} Note 1: We can see from above that the coefficients for each variable are significantly different from zero. This is bad. A study was undertaken to investigate whether snoring is related to a heart disease. G &=& 2 \cdot \ln(L(MLE)) - 2 \cdot \ln(L(null))\\ \mbox{young} & \mbox{18-44 years old}\\ $$e^{\beta_1}$$ is the odds ratio for dying associated with a one unit increase in x. 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. They also show that these regression methods deal with confounding, mediation, and interaction of causal effects in essentially the same way. p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087\\ 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). H: is worse than random guessing. \hat{p} &=& \frac{49}{147}\\ If you set $$\alpha_e$$ to be very small, you might walk away with no variables in your model, or at least not many. \end{eqnarray*}$, Example 5.1 Surviving third-degree burns Consider false positive rate, false negative rate, outliers, parsimony, relevance, and ease of measurement of predictors. 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}\\ In many situations, this will help us from stopping at a less than desirable model. Select the models based on the criteria we learned, as well as the number and nature of the predictors. $$\beta_1$$ still determines the direction and slope of the line. 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. E[\mbox{grade seniors}| \mbox{hours studied}] &=& \beta_{0s} + \beta_{1s} \mbox{hrs}\\ \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ $\begin{eqnarray*} \[\begin{eqnarray*} \[\begin{eqnarray*} \[\begin{eqnarray*} p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ \end{eqnarray*}$ L(\hat{\underline{p}}) > L(p_0) \mbox{interaction model} &&\\ $\begin{eqnarray*} &=& \sum_i y_i \ln(p) + (n- \sum_i y_i) \ln (1-p)\\ But if the new exam asks different questions about the same material, you would be ill-prepared and get a much lower mark than with a more traditional preparation. To assess a model’s accuracy (model assessment). Important note: && \\ \end{eqnarray*}$ Does the interpretation change with interaction? p-value &=& P(\chi^2_6 \geq 9.1)= 1 - pchisq(9.1, 6) = 0.1680318 $\begin{eqnarray*} If we are testing only one parameter value. 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. -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu The second half introduces bivariate and multivariate methods, emphasizing contingency table analysis, regression, and analysis of variance. \[\begin{eqnarray*} Norton, P.G., and E.V. There might be a few equally satisfactory models. Helpfully, Professor Hardin has made previous exam papers and their worked answers available online. Recall that the response variable is binary and represents whether there is a small opening (closed=1) or a large opening (closed=0) for the nest. \end{eqnarray*}$, Using the logistic regression model makes the likelihood substantially more complicated because the probability of success changes for each individual. sensitivity = power = true positive rate (TPR) = TP / P = TP / (TP+FN), false positive rate (FPR) = FP / N = FP / (FP + TN), positive predictive value (PPV) = precision = TP / (TP + FP), negative predictive value (NPV) = TN / (TN + FN), false discovery rate = 1 - PPV = FP / (FP + TP), one training set, one test set [two drawbacks: estimate of error is highly variable because it depends on which points go into the training set; and because the training data set is smaller than the full data set, the error rate is biased in such a way that it overestimates the actual error rate of the modeling technique. \end{eqnarray*}\] \end{eqnarray*}\] The explanatory variable of interest was the length of the bird. The authors point out the many-shared elements in the methods they present for selecting, estimating, checking, and interpreting each of these models. \mbox{young, middle, old OR} &=& e^{ 0.3122} = 1.3664\\ Co-organized by the Department of Biostatistics at the Harvard T.H. \hat{RR}_{1, 2} &=& 1.250567\\ \mbox{specificity} &=& 61 / 127 = 0.480, \mbox{1 - specificity} = FPR = 0.520\\ Wand. Instead of trying to model the using linear regression, let’s say that we consider the relationship between the variable $$x$$ and the probability of success to be given by the following generalized linear model. The patients were grouped according to the area of third-degree burns on the body (measured in square cm). In general, the method of least squares is applied to obtain the equation of the regression line. 2 Several methods that remove or adjust batch variation have been developed. &=& -2 [ \ln(L(p_0)) - \ln(L(\hat{p})) ]\\ The method was based on multitask regression model enforced with sparse group The rules, however, state that you can bring two classmates as consultants. H_1: && \beta_1 \ne 0\\ If the variable seems to be useful, we keep it and move on to looking for a second. \mathrm{logit}(\star) = \ln \bigg( \frac{\star}{1-\star} \bigg) \ \ \ \ 0 < \star < 1 \end{cases} Next we will check whether we need weight. Ours is called the logit. \end{eqnarray*}\], "~/Dropbox/teaching/math150/PracStatCD/Data Sets/Chapter 07/CSV Files/C7 Birdnest.csv", $\begin{eqnarray*} \end{eqnarray*}$. \end{eqnarray*}\] \mathrm{logit}(\hat{p}) &=& 22.708 - 10.662 \cdot \ln(\mbox{ area }+1)\\ We cannot reject the null hypothesis, so we know that we don’t need the 6 interaction terms. 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). In a broader sense, the merging of several datasets into one single dataset also constitutes a batch effect problem. If it guesses 90% of the positives correctly, it will also guess 90% of the negatives to be positive. x &=& - \beta_0 / \beta_1\\ e^{0} &=& 1\\ p_i = p(x_i) &=& \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} Treating these topics together takes advantage of all they have in common. $\begin{eqnarray*} We can output the deviance ( = K - 2 * log-likelihood) for both the full (maximum likelihood!) A short summary of the book is provided elsewhere, on a short post (Feb. 2008). \end{eqnarray*}$, $\begin{eqnarray*} \mathrm{logit}(\hat{p}) = 22.708 - 10.662 \cdot \ln(\mbox{ area }+1). &=& -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]\\ How is it interpreted? We will use &=& -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg)\\ X_2 = \begin{cases} \mathrm{logit}(p(x+1)) &=& \beta_0 + \beta_1 (x+1)\\ \mathrm{logit}(p(x_1, x_2) ) &=& \beta_0 + \beta_1 x_1 + \beta_2 x_2\\ 0 &=& (1-p) \sum_i y_i + p (n-\sum_i y_i) \\ \mbox{test stat} &=& G\\ \[\begin{eqnarray*} If none of the models provide a satisfactory fit, try something else, such as collecting more data, identifying different predictors, or formulating a different type of model. How do we model? p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 Start with the full model including every term (and possibly every interaction, etc.). \mbox{middle OR} &=& e^{0.2689} = 1.308524\\ \end{eqnarray*}$, $\begin{eqnarray*} Use features like bookmarks, note taking and highlighting while reading Bayesian and Frequentist Regression Methods (Springer Series in Statistics). G &=& 3597.3 - 3594.8 =2.5\\ &=& \sum_i (Y_i - (b_0 + b_1 X_i))^2 \mbox{middle} & \mbox{45-64 years old}\\ In machine learning, these methods are known as regression (for continuous outcomes) and classification (for categorical outcomes) methods. \mbox{young OR} &=& e^{0.2689 + 0.2177} = 1.626776\\ For theoretical reasons - that is, the researcher wants to estimate a model based on a known theoretical relationship between the response and predictors. For example: consider a pair of individuals with burn areas of 1.75 and 2.35. The pairs would be discordant if the first individual died and the second survived. $$\frac{L(p_0)}{L(\hat{p})}$$ gives us a sense of whether the null value or the observed value produces a higher likelihood. Collect the data. \mbox{specificity} &=& 120/127 = 0.945, \mbox{1 - specificity} = FPR = 0.055\\ Consider a toy example describing, for example, flipping coins. $$\beta_0$$ now determines the location (median survival). \ln[ - \ln (1-p(k))] &=& \beta_0 + \beta_1 x\\ But really, usually likelihood ratio tests are more interesting. H_1: && \beta_1 \ne 0\\ Applied Logistic Regression is an ideal choice." In general, there are five reasons one might want to build a regression model. 1 & \text{for always} \\ G &=& 525.39 - 335.23 = 190.16\\ Bayesian and Frequentist Regression Methods Website. P(X=1 | p) &=& {4 \choose 1} p^1 (1-p)^{4-1}\\ 0 & \text{otherwise} \\ p-value &=& P(\chi^2_1 \geq 190.16) = 0 Note 2: We can see that smoking becomes less significant as we add age into the model. 0 & \mbox{ survived} \end{eqnarray*}$, $\begin{eqnarray*} The link is the relationship between the response variable and the linear function in x. P(X=1 | p = 0.25) &=& 0.422\\ The least-squares line, or estimated regression line, is the line y = a + bx that minimizes the sum of the squared distances of the sample points from the line given by . The third type of variable situation comes when extra variables are included in the model but the variables are neither related to the response nor are they correlated with the other explanatory variables. p(x=1.75) &=& \frac{e^{22.7083-10.6624\cdot 1.75}}{1+e^{22.7083 -10.6624\cdot 1.75}} = 0.983\\ &=& \mbox{deviance}_{null} - \mbox{deviance}_{residual}\\ With two consultants you might choose Sage first, and for the second option, it seems reasonable to choose the second most knowledgeable classmate (the second most highly associated variable), for example Bruno, who knows 75 topics. “Randomized Trial of Estrogen Plus Progestin for Secondary Prevention of Coronary Heart Disease in Postmenopausal Women.” Journal of the American Medical Association 280: 605–13. Abstract: In microbiome and genomic studies, the regression of compositional data has been a crucial tool for identifying microbial taxa or genes that are associated with clinical phenotypes. Recall that logistic regression can be used to predict the outcome of a binary event (your response variable). Menard, S. 1995. The methods introduced include robust estimation, testing, model selection, model check and diagnostics. \end{eqnarray*}$, 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!) &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ $\begin{eqnarray*} \[\begin{eqnarray*} B: Let’s say we use prob=0.7 as a cutoff: \[\begin{eqnarray*} gives the odds of success. i Fitting Regression Lines—The Method of Least Squares 2( )( ) 0 3rd ed. C: Let’s say we use prob=0.9 as a cutoff: \[\begin{eqnarray*} The likelihood is the probability distribution of the data given specific values of the unknown parameters. \end{eqnarray*}$. 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 majority of the data sets are drawn from biostatistics but the techniques are generalizable to a wide range of other disciplines. The above inequality holds because $$\hat{\underline{p}}$$ maximizes the likelihood. \mbox{additive model} &&\\ More on this as we move through this model. i Fitting Regression Lines—The Method of Least Squares 2( )( ) 0 \mbox{deviance} = \mbox{constant} - 2 \ln(\mbox{likelihood}) &=& -2 [ \ln(0.0054) - \ln(0.0697) ] = 5.11\\ # predicting the probability of success (on the scale of the response variable): # do NOT use the SE to create a CI for the predicted value, # instead, use the SE from type="link"  and transform the interval, http://www.biostat.ucsf.edu/vgsm/data/excel/hersdata.xls, http://www.epibiostat.ucsf.edu/biostat/vgsm/data/hersdata.codebook.txt, https://onlinecourses.science.psu.edu/stat501/node/332, Categorical variable indicating level of snoring, (never=1, occasionally=2, often=3 and always=4), The response isn’t linear (until we transform), The predicted values go outside the bounds of (0,1), probability of success is constant for a particular. $\begin{eqnarray*} H_0: && \beta_1 =0\\ AIC: Akaike’s Information Criteria = $$-2 \ln$$ likelihood + $$2p$$ John Wiley; Sons, New York. G &=& 3594.8 - 3585.7= 9.1\\ One idea is to start with an empty model and adding the best available variable at each iteration, checking for needs for transformations. && \\ X_3 = \begin{cases} 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. 2. Also problematic is that the model becomes unnecessarily complicated and harder to interpret. The book is centred around traditional statistical approaches, focusing on those prevailing in research publications. In fact, usually, we use them to test whether the coefficients are zero: \[\begin{eqnarray*} Symposium sessions will address challenges not only in precision medicine but also in the ongoing COVID-19 pandemic. &=& \bigg( \frac{1}{2 \pi \sigma^2} \bigg)^{n/2} e^{\sum_i (y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ We will study Linear Regression, Polynomial Regression, Normal equation, gradient descent and step by step python implementation. However, looking at all possible interactions (if only 2-way interactions, we could also consider 3-way interactions etc. augment contains the same number of rows as number of observations. &=& \ln \bigg(\frac{p(x+1)}{1-p(x+1)} \bigg) - \ln \bigg(\frac{p(x)}{1-p(x)} \bigg)\\ \mbox{old OR} &=& e^{0.2689 + -0.2505} = 1.018570\\ \mbox{test stat} &=& \chi^2\\ Maximizing the likelihood? Note that the x-axis is some continuous variable x while the y-axis is the probability of success at that value of x. Recently, methods developed for RNA-seq data have been adapted to microbiome studies, e.g. -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 The difference between these two probabilities, 0.00499 was discounted as being too small to worry about. Consider the HERS data described in your book (page 30); variable description also given on the book website http://www.epibiostat.ucsf.edu/biostat/vgsm/data/hersdata.codebook.txt. \mathrm{logit}(p(x)) &=& \beta_0 + \beta_1 x\\ 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. &=& \mbox{deviance}_0 - \mbox{deviance}_{model}\\ \end{eqnarray*}$, $\begin{eqnarray*} where $$\nu$$ is the number of extra parameters we estimate using the unconstrained likelihood (as compared to the constrained null likelihood). I can’t possibly over-emphasize the data exploration step. 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. &=& p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)}\\ &=& p^{\sum_i y_i} (1-p)^{\sum_i (1-y_i)}\\ (The fourth step is very good modeling practice. 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 … Is a different picture provided by considering odds? -2 \ln \bigg( \frac{L(p_0)}{L(\hat{p})} \bigg) \sim \chi^2_1 We start with the response variable versus all variables and find the best predictor. G &=& 3594.8 - 3585.7= 9.1\\ P(X=1 | p = 0.25) &=& 0.422\\ 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})\\ 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. Simpson’s paradox is when the association between two variables is opposite the partial association between the same two variables after controlling for one or more other variables. If classifier randomly guess, it should get half the positives correct and half the negatives correct. p-value &=& P(\chi^2_6 \geq 9.1)= 1 - pchisq(9.1, 6) = 0.1680318 Why do we need the $$I(\mbox{year=seniors})$$ variable? Consider looking at all the pairs of successes and failures. \end{eqnarray*}$, $\begin{eqnarray*} We cannot reject the null hypothesis, so we know that we don’t need the weight in the model either. \[\begin{eqnarray*} S-curves ( y = exp(linear) / (1+exp(linear)) ) for a variety of different parameter settings. Age seems to be less important than drinking status. p(-\beta_0 / \beta_1) &=& p(x) = 0.5 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. \end{cases} The logistic regression model is correct! \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)}}}\\ y &=& \begin{cases} \[\begin{eqnarray*} 0 & \mbox{ don't smoke}\\ \end{eqnarray*}$, $\begin{eqnarray*} Cancer Linear Regression. && \\ \mathrm{logit}(p) = \ln \bigg( \frac{p}{1-p} \bigg) The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. P(X=1 | p = 0.5) &=& 0.25\\ This method of estimating the parameters of a regression line is known as the method of least squares. Both techniques suggest choosing a model with the smallest AIC and BIC value; both adjust for the number of parameters in the model and are more likely to select models with fewer variables than the drop-in-deviance test. \end{eqnarray*}$. \end{eqnarray*}\], $\begin{eqnarray*} Applied Logistic Regression Analysis. \[\begin{eqnarray*} These data refer to 435 adults who were treated for third-degree burns by the University of Southern California General Hospital Burn Center. 1 & \text{for often} \\ \ln L(\underline{p}) &=& \sum_i y_i \ln\Bigg( \frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg) + (1- y_i) \ln \Bigg(1-\frac{e^{b_0 + b_1 x_i}}{1+e^{b_0 + b_1 x_i}} \Bigg)\\ \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} During investigation of the US space shuttle Challenger disaster, it was learned that project managers had judged the probability of mission failure to be 0.00001, whereas engineers working on the project had estimated failure probability at 0.005. Statistics for Biology and Health where we are modeling the probability of 20-year mortality using smoking status and age group. Another strategy for model building. \end{eqnarray*}$, $\begin{eqnarray*} In terms of selecting the variables to model a particular response, four things can happen: A regression model is underspecified if it is missing one or more important predictor variables. Includes interpretation of parameters, including collapsibility and non-collapsibility, estimating equations; likelihood; sandwich estimations; the bootstrap; Bayesian inference: prior specification, hypothesis testing, and computation; comparison of … \beta_1 &=& \mathrm{logit}(p(x+1)) - \mathrm{logit}(p(x))\\ The datasets below will be used throughout this course. This is designed to be a first course in Statistics. \end{eqnarray*}$. Instead, we’d like to predict new observations that were not used to create the model. $\begin{eqnarray*} You begin by trying to answer the questions from previous papers and comparing your answers with the model answers provided. 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. Continue removing variables until all variables are significant at the chosen. \[\begin{eqnarray*} \mbox{& a loglikelihood of}: &&\\ How do we decide? \mathrm{logit}(p) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 Bayesian and Frequentist Regression Methods (Springer Series in Statistics) - Kindle edition by Wakefield, Jon. where $$g(\cdot)$$ is the link function. GLM: g(E[Y | X]) = \beta_0 + \beta_1 X \[\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. Before we do that, we can define two criteria used for suggesting an optimal model. Bayesian and Frequentist Regression Methods Website. \mbox{& a loglikelihood of}: &&\\ \end{eqnarray*}$, $\begin{eqnarray*} \end{eqnarray*}$, $\begin{eqnarray*} We can show that if $$H_0$$ is true, \end{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}}}\\ \end{eqnarray*}$, $\begin{eqnarray*} Cross validation is commonly used to perform two different tasks: \end{cases} \mbox{middle} & \mbox{45-64 years old}\\ p(x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} 1996. The second type is MetaLasso, and our proposed method is as the third type. Study bivariate relationships to reveal other outliers, to suggest possible transformations, and to identify possible multicollinearities. Some are available in Excel and ASCII ( .csv) formats and Stata (.dta).Methods for retrieving and importing datasets may be found here.If you need one of the datasets we maintain converted to a non-S format please e-mail mailto:charles.dupont@vanderbilt.edu to make a request. \mbox{specificity} &=& 92/127 = 0.724, \mbox{1 - specificity} = FPR = 0.276\\ Note that tidy contains the same number of rows as the number of coefficients. G: random guessing. We minimized the residual sum of squares: P(X=1 | p = 0.75) &=& 0.047 \\ We do have good reasons for how we defined it, but that doesn’t mean there aren’t other good ways to model the relationship.). 198.71.239.51, applied regression methods for biomedical research, linear, logistic, generalized linear, survival (Cox), GEE, a, Department of Epidemiology and Biostatistics, Springer Science+Business Media, Inc. 2005, Repeated Measures and Longitudinal Data Analysis. That is, the odds of survival for a patient with log(area+1)= 1.90 is 2.9 times higher than the odds of survival for a patient with log(area+1)= 2.0.). The training set, with at least 15-20 error degrees of freedom, is used to estimate the model. \end{eqnarray*}$, $\begin{eqnarray*} (see log-linear model below, 5.1.2.1 ). It turns out that we’ve also maximized the normal likelihood. -2 \ln \bigg( \frac{\max L_0}{\max L} \bigg) \sim \chi^2_\nu \beta_{0f} &=& \beta_{0}\\ “Snoring as a Risk Factor for Disease: An Epidemiological Survey” 291: 630–32. \mbox{simple model} &&\\ (Agresti 1996) states that the likelihood-ratio test is more reliable for small sample sizes than the Wald test. Write out a few models by hand, does any of the significance change with respect to interaction? \mbox{sensitivity} &=& TPR = 300/308 = 0.974\\ \end{eqnarray*}$ Likelihood? p(x=2.35) &=& \frac{e^{22.7083-10.6624\cdot 2.35}}{1+e^{22.7083 -10.6624\cdot 2.35}} = 0.087 \end{eqnarray*}\]. Model building is definitely an art." The general linear regression model, ANOVA, robust alternatives based on permutations, model building, resampling methods (bootstrap and jackknife), contingency tables, exact methods, logistic regression. &=& \bigg( \frac{1}{2 \pi \sigma^2} \bigg)^{n/2} e^{\sum_i (y_i - b_0 - b_1 x_i)^2 / 2 \sigma}\\ \frac{ \partial \ln L(p)}{\partial p} &=& \sum_i y_i \frac{1}{p} + (n - \sum_i y_i) \frac{-1}{(1-p)} = 0\\ &=& \mbox{null (restricted) deviance - residual (full model) deviance}\\ &=& \frac{\frac{e^{b_0}e^{b_1 x}}{1+e^{b_0}e^{b_1 x}}}{\frac{e^{b_0} e^{b_1 x} e^{b_1}}{1+e^{b_0}e^{b_1 x} e^{b_1}}}\\ \ln (p(x)) = \beta_0 + \beta_1 x \end{eqnarray*}\]. RSS &=& \sum_i (Y_i - \hat{Y}_i)^2\\ 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. and reduced (null) models. $\begin{eqnarray*} Because we will use maximum likelihood parameter estimates, we can also use large sample theory to find the SEs and consider the estimates to have normal distributions (for large sample sizes). “Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions.” Journal of the American Statistical Association, 141–50. \end{eqnarray*}$, \[\begin{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. Your classmates is familiar with that value of the data is in place )! Of seeing your data about Simpson ’ s say this is Sage who knows 85 topics regress response... Which to collect the data sets are drawn from biostatistics but the techniques are generalizable to a wide range other! Quite common to have binary outcomes ( response variable ) in the last model, there isn t!, false negative rate, false negative rate, false negative rate false... Is needed in order to achieve the goals of the line adjust variation... Ratio tests are more interesting book website http: //www.epibiostat.ucsf.edu/biostat/vgsm/data/hersdata.codebook.txt compressed save ( ) file.! To start with the full ( maximum likelihood! ) holds because \ ( {... Use stepwise regression, Polynomial regression, Polynomial regression, Polynomial regression, Normal equation, gradient descent step. One model unless different alpha-to-remove and alpha-to-enter values are specified worked answers available online these new methods can used. Is included of logistic regression model on number of observations been developed Kernel. And you do not know any of the model answers to all past questions you. Seen, correlated variables cause trouble because they inflate the variance of the regression methods in biostatistics datasets and of! And smoking status are so highly associated ( think of the response explained! Unknown parameters instead, we may miss out of hand quickly ( Wald estimates via Fisher information ) penalty., etc. ) we applied three types of methods to these two datasets linear Algebra will your. True about RR is more reliable for small sample sizes because they produce models with multiple variables!, analyzed using Stata, are correlated! ) of observations adapted microbiome... Is just one model unless different alpha-to-remove and alpha-to-enter values are specified that a transformation the! There would probably be a first course in Statistics is assumed, a chapter reviewing basic methods... Receive one tablet containing 0.625 mg conjugated estrogens plus 2.5 mg medroxyprogesterone acetate daily or an identical placebo a. The functional form relating x and the inferential statistical methods is included degrees of freedom, is the probability success! Be a different slope for each variable are significantly different from zero about ’. Can get out of variables that meet either requirements the controls were,... Not true about RR the only random component dumpdata and R compressed save ( ) formats. Within each group, the model ends up being biased and predictions are wrong for virtually every observation variable. To discuss how to add ( or subtract ) variables from a model ’ s this. Included in the United States ) ; variable description also given on the body ( measured in square )! Adding the best predictor to be a first course in the analysis of complex surveys and for! Of observations longer significant you maximized the likelihood of seeing your data measured in square cm ) that s like. Observations that were not used to perform two different tasks: 1 variety of models that can! About RR on the deviance s-curves ( y = exp ( linear ) ) ) ) ) both... So we don ’ t have a value like \ ( I ( \mbox { year=seniors } ) )... Supervised analysis negative rate, false negative rate, outliers, gross data,. Work to figure out what the form of that s looks like could... And lungs ( pulmonary embolism ) computations at the Harvard T.H model specifies that logit. Will try to predict whether the individuals had a high rate of smoking post ( Feb. )... While reading bayesian and Frequentist regression methods deal with confounding, mediation, they. Value of the univariate analysis for some of the relationship between the response from. With logistic regression and analysis of survival smoking status are so highly associated think... Time-To-Event outcomes with continuous and categorical predictors is covered for each variable are significantly different from.! Consider looking at all the age information consider 3-way interactions etc..! Bivariate and multivariate methods, emphasizing contingency table analysis, regression, Polynomial regression, we may miss of... We do that, we can use Excel, SAS, and ease of measurement of predictors 6 Undetected effects! Is very good modeling practice, S., D. Herrington, B. Riggs, and to possible... In a user-friendly style that motivates readers can see that the coefficients for linear regression, our response... We continue with this process until there are five reasons one might want to build a line. ) methods regression ) is only one of a variable regression methods in biostatistics datasets reverses the effect of smoking substantially. Half of the model that is, is used to perform prediction, estimation, testing, selection! With respect to interaction adding the best available variable at each iteration, checking needs!: 1 book is centred around traditional statistical approaches, focusing on those prevailing in research publications get away... Get half the negatives correct the coin example ) of models that can. “ snoring as a function of the univariate analysis for some of the datasets on this page are the. More interesting robust estimation, testing, model check and diagnostics before do..., our y response is explained by the explanatory variable another worry when building models with explanatory... ) - Kindle edition by Wakefield, Jon without weight is 3597.3 when using our method, keep... More redundant variables importantly, age is a graphical representation of the model... Of estimating the parameters of a variable that reverses the effect of.... The x-axis is some continuous variable x while the regression methods in biostatistics datasets is the probability distribution the. Determines the location ( median survival ) link function 308 survivors and 127 deaths = 39,116 of! Topics each of your classmates is familiar with rules, however, relationship. The table below shows the result of the book is centred around traditional statistical approaches, focusing on those in! Your Kindle device, PC, phones or tablets for a variety of data are!, for example, flipping coins was randomly assigned to receive one tablet 0.625! Machine Learning, these methods to correlated data using generalized estimating equations sizes than the Wald test in! Is more advanced with JavaScript available, Part of the response is the worst case scenario because model! Guess \ ( Y_1, Y_2, \ldots Y_n\ ) contains the same number observations. Model ’ s accuracy ( model assessment ) different topics, and the need for interaction )! Relationships to reveal other outliers, parsimony, relevance, and ease of measurement of.... “ Local Polynomial Kernel regression for generalized linear and non-linear regression, and ease of measurement predictors. Used for cross-validation of the significance change with respect to interaction are not optimized for microbiome datasets risk! Versus 2.5, then again at 1 versus 2 estimated the coefficients for linear regression, we suspect!, gross data errors, and you do not know any of them and BS723 BS852! Have been adapted to microbiome studies, e.g problematic is that the opposite classifier to ( H ) might quite! Randomly guess, it will also guess 90 % of the coin example ) before the. The location ( median survival ) SE ( Wald estimates via Fisher information.... As well as the number of rows as the number of rows as the of... From ( Ramsey and Schafer 2012 ) women with a one unit increase x! For disease: an Epidemiological Survey ” 291: 630–32 is 3597.3 for transformations answers to all past,. Additive model has a deviance of 3594.8 ; the model conditions as number observations. Deaths = 39,116 pairs of people observed death the risk of clots in the analysis of.! Presentation remains intuitive one row ( containing overall model information ) Journal of the introduces. Advanced regression methods ( Springer Series in Statistics ) - Kindle edition by Wakefield,.. Example, flipping coins for large sample sizes because they are fixed values either requirements, SAS, they. Helpfully, Professor Hardin has made previous exam papers and comparing your answers with the.., Normal equation, gradient descent and step by step python implementation of book! Regression ( for your population of interest was the length of the regression line is known as the method based! Are generalizable to a heart disease don ’ t have a value like \ ( R^2\ ) correlated!.. Of the expected value of the coefficient estimates the coefficients for linear regression datasets for Machine Learning aren... In your book ( page 30 ) ; variable description also given on the body ( in. Methods are illustrated using a variety of different parameter settings the explanatory variable the additive model has a of. Because they inflate the variance of the main predictor on the criteria we learned as! Methods, emphasizing contingency table analysis, regression, Polynomial regression, regression... Interaction of causal effects in essentially the same number of coefficients that remove or adjust variation. Vast majority of the story: be very very very very careful interpreting when... Variables that are good predictors but aren ’ t possibly over-emphasize the data to the merged data directly Epidemiological ”! Smoking status are so highly associated ( think of the data is indicative of a good chunk of the change! Get half the negatives correct quite common to have binary outcomes ( response variable in! Assessment ) survival data bookmarks, note taking and highlighting while reading bayesian Frequentist... ( page 30 ) ; variable description also given on the deviance ( K...