Introduction to linear model

Transcription

1 Introduction to linear model Valeska Andreozzi 2012 References 3 Correlation 5 Definition Pearson correlation coefficient Spearman correlation coefficient Hypothesis test Simple linear regression 15 Motivation The model Model assumptions Fitting the model Exercise Multiple linear regression 29 The model Hypothesis test Variable selection Model check Interaction

2 Summary References Correlation Definition Pearson correlation coefficient Spearman correlation coefficient Hypothesis test Simple linear regression Motivation The model Model assumptions Fitting the model Exercise Multiple linear regression The model Hypothesis test Variable selection Model check Interaction DEIO/CEAUL Valeska Andreozzi slide 2 References slide 3 References Rosner, B (2010). Fundamentals of Biostatistics. 7 th Edition. Duxbury Resource Center. Krzanowski, W (1998). An Introduction to Statistical Modelling. Arnold Texts in Statistics. Harrel, F (2001). Regression Modeling Strategies. Springer-Verlag. Weisberg, S (2005). Applied Linear Regression (Wiley Series in Probability and Statistics). Third Edition. Wiley. Dalgaard, P (2008). Introductory Statistics with R (Statistics and Computing). Second Edition. Springer. DEIO/CEAUL Valeska Andreozzi slide 4 Correlation slide 5 Definition The correlation coefficient is a dimensionless quantity that is independent of the units of the random variables X and Y and ranges between 1 and 1 For random variables that are approximately linearly related, a correlation coefficient of 0 implies independence A correlation coefficient close to 1 implies nearly perfect positive dependence with large values of X corresponding to large values of Y and small values of X corresponding to small values of Y A correlation coefficient close to 1 implies perfect negative dependence with large values of X corresponding to small values of Y and vice versa DEIO/CEAUL Valeska Andreozzi slide 6 2

3 Examples An example of a strong positive correlation is between forced respiratory volume (FEV) and height. A weaker positive correlation exist between serum cholesterol and dietary intake cholesterol. A strong negative correlation can be found between resting pulse and age in children under the age of 10y. DEIO/CEAUL Valeska Andreozzi slide 7 Pearson correlation coefficient Assuming that Y e X are two random variables with a linear relationship, we can measure the correlation of a sample calculating the Pearson correlation coeficient, given by: i r = (x i x)(y i ȳ) i (x i x) 2 i (y i ȳ) 2 DEIO/CEAUL Valeska Andreozzi slide 8 in R library(iswr) data(thuesen) attach(thuesen) View(thuesen) plot(thuesen) cor(blood.glucose, short.velocity) cor(blood.glucose, short.velocity,use="complete.obs") Correlation in R Commander Statistics > Summaries > Correlation matrix... DEIO/CEAUL Valeska Andreozzi slide 9 3

4 Spearman correlation coefficient Adequate when one or both variables are either ordinal or have a distribution that is far from normal The Spearman correlation coefficient is a nonparametric method which has the advantage of being invariant to monotone transformation of the coordinates. The man disadvantage of this method is that its interpretation is not quite clear. The Spearman (rank) correlation coefficient r s is obtained by replacing the observation of X and Y by their ranks and computing the correlation (Pearson coefficient). It is assumed that if there were a perfect correlation between two variables, then the ranks for each person on each variable would be the same and r s = 1. The less perfect the correlation, the closer to zero r s would be. DEIO/CEAUL Valeska Andreozzi slide 10 in R Correlation in R x<-rank(thuesen$blood.glucose[-16]) y<-rank(thuesen$short.velocity[-16]) cor(x,y,method="pearson") cor(thuesen$blood.glucose,thuesen$short.velocity, use="complete.obs",method="spearman") Correlation in R Commander Statistics > Summaries > Correlation matrix... DEIO/CEAUL Valeska Andreozzi slide 11 Hypothesis test Is is possible to test the significance of the correlation by tranforming it to a t-distributed variable, which will be identical with the test obtained from testing the significance of the slope of either regression of y on x, or vice-versa (see later). In R cor.test(blood.glucose, short.velocity) in R Commander Statistics > Summaries > Correlation test... DEIO/CEAUL Valeska Andreozzi slide 12 4

5 Exercises Match the following items with graphics. r = 0 0 < r < 1 r = 1 r = 1 1 < r < 0 DEIO/CEAUL Valeska Andreozzi slide 13 Exercises DEIO/CEAUL Valeska Andreozzi slide 14 5

6 Simple linear regression slide 15 Motivation What are the relationship between sistolic blood pressure (SBP) and age among health adults? SBP increases with age There are fluctuations around a linear trend Variability of SBP not completely explained by age random component Why would we like to fit a model? Describe the relationship between SBP and age Prediction DEIO/CEAUL Valeska Andreozzi slide 16 Motivation What can we say about SBP age? pa id DEIO/CEAUL Valeska Andreozzi slide 17 6

7 General concepts y i = β 0 + β 1 x i + ǫ i pa Positive linear correlation Relationship is not perfect. Fitted line which describe the linear relationship between SBP and age. ŷ i = x i id DEIO/CEAUL Valeska Andreozzi slide 18 Model interpretation ŜBP i = age i pa ˆβ 0 = estimated value of SBP when age is zero ˆβ 1 = 0.97 the SPB increases 0.97 mmhg for an increment on one year of age id DEIO/CEAUL Valeska Andreozzi slide 19 7

8 Model illustration Illustration of the components of a simple linear regression model. Systematic component: β 0 + β 1 x i Statistical/Probabilistic Model: Y i = β 0 + β 1 x i + ǫ i or E(Y i ) = β 0 + β 1 x i DEIO/CEAUL Valeska Andreozzi slide 20 Model illustration Simple linear regression representation. The means of the probability distributions of Y i show the sistematic relation with X DEIO/CEAUL Valeska Andreozzi slide 21 Model assumptions Independence: Y i are all independent Linearity: The expected value of Y i is a linear function of X i Homogeneity of variance: The variance of Y i probability distribution is constant over X and equal to σ 2 Normality: For all X i, Y i follows a Normal distribution. Assumption necessary to build hypothesis test and confidence intervals of the model parameters β DEIO/CEAUL Valeska Andreozzi slide 22 8

9 Model estimation Least Squares Method E(Y i X) = β 0 + β 1 x i LSM gives estimates β 0 and β 1 that minimize the sum of squared errors (SSE) SSE = = n i=1 ǫ 2 i n (y i ŷ i ) 2 = i=1 i=1 n (y i β 0 β 1 x i ) 2 DEIO/CEAUL Valeska Andreozzi slide 23 Model estimation β coefficients Differentiating SSE and setting the partial derivatives to zero, we have: SSE β 0 = SSE β 1 = n [y i β 0 β 1 x i ] = 0 i=1 n [x i (y i β 0 β 1 x i )] = 0 i=1 The system results give the estimates of the model parameters ˆβ 0 = ȳ ˆβ 1 x n i=1 ˆβ 1 = (x i x)(y i ȳ) n i=1 (x i x) 2 DEIO/CEAUL Valeska Andreozzi slide 24 Model estimation Variance of Y (σ 2 ) Under the null hypothesis that the residuals are independent random variables with zero mean and variance constant equal to σ 2, an unbiased estimator for σ 2 is the ratio between SSE = n i=1 ǫ2 i and the degree of freedom (the number of observation minus the number of model coefficients) And then, the variance σ 2 of Y is obtained. DEIO/CEAUL Valeska Andreozzi slide 25 9

10 in R Simple linear regression in R dados<-read.table("pasis.dat",header=t) names(dados) head(dados) plot(dados) modelo<-lm(pa~id,data=dados) summary(modelo) plot(dados) abline(modelo,col=2) DEIO/CEAUL Valeska Andreozzi slide 26 in R Commander Simple linear regression in R Commander Data > Import data > fromm text file, clipboard, or URL,... Graphics > Scatterplot... Statistics > Fit Models > Linear regression... DEIO/CEAUL Valeska Andreozzi slide 27 Exercise Exercise in R 1. With the rmr dataset (ISwR package), plot the metabolic rate versus body weight. Fit a linear regression model to the relation. According to the fitted model, what is the predicted metabolic rate for a body weight of 70kg? 2. In the jull dataset (ISwR package) fit a linear regression model to the square root of the IGF-I concentration versus age, to the group of subjects over 25 years old. Tools > Load package(s)... Data > Data in packages > Read data set from an attached package... Graphics > Scatterplot... Statistics > Fit Models > Linear regression... Data > Manage variable in active data set > Compute new variable... DEIO/CEAUL Valeska Andreozzi slide 28 10

11 Multiple linear regression slide 29 Multiple linear regression y i = β 0 + β 1 x 1i + β 2 x 2i + ǫ i Describe the relationship between the response (dependent) variable (outcome) (Y ) and two or more independent variables (covariates, predictors, explanatory variables) (X 1,X 2,X 3,,X k ) Estimate the direction of the association between response variable and covariates. The covariates can be transformed variables (example: log(cd4)), polynomials terms (example: age 2 ), interaction terms (example age sex) and dummy variables. Determinate which covariates are important to predict the response variable Describe the relationship of X 1,X 2,X 3,,X k and Y adjusted by the effect of other covariates Z 1 and Z 2, for example. DEIO/CEAUL Valeska Andreozzi slide 30 Multiple linear regression y i = β 0 + β 1 x 1i + β 2 x 2i + ǫ i Assumes that the response variable is an random variable which varies from individual to individual i. The nature of the continuous response variable suggests that the Normal distribution is adequate to the population model of Y i So, Y i follows a Normal distribution with mean µ i and variance σ 2 unknown. (Y i N(µ i,σ 2 )) Similarly, we can say that each observation y i = µ i + ǫ i and that ǫ i N(0,σ 2 ) The model parameters are also estimated by least square method. DEIO/CEAUL Valeska Andreozzi slide 31 11

12 Exemplo Describe the relationship between blood preassure (y i ) and age (x 1i ), body mass index (x 2i ) and smoke habits (x 3i ). File: (multi.dat) E(Y i ) = β 0 + β 1 x 1i + β 2 x 2i + β 3 x 3i Data > Import data > from a text file, clipboard or URL... Statistics > Summaries > Active data set summary(dados) pessoa pa id Min. : 1.00 Min. :120.0 Min. : st Qu.: st Qu.: st Qu.:48.00 Median :16.50 Median :143.0 Median :53.50 Mean :16.50 Mean :144.5 Mean : rd Qu.: rd Qu.: rd Qu.:58.25 Max. :32.00 Max. :180.0 Max. :65.00 imc hf Min. :2368 n~ao:15 1st Qu.:3022 sim:17 Median :3380 Mean :3441 3rd Qu.:3776 Max. :4637 DEIO/CEAUL Valeska Andreozzi slide 32 in R Commander Statitics > Fit models > Linear models... Call: lm(formula = pa ~ id + imc + hf, data = multi) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) *** id *** imc hf[t.sim] *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 28 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 3 and 28 DF, p-value: 7.602e-09 DEIO/CEAUL Valeska Andreozzi slide 33 12

13 Model interpretation β 0 = The intercept has no real interpretation in the example because is the average BP of a person with age zero, body mass index equal to zero and who smokes. β 1 = The BP increases, on average, 1.21 mmhg for an increase of 1 year of age adjusted by body mass index and smoke habits β 2 = The BP increases, on average, mmhg for an increase of 1 unit of body mass index, holding everything else constant β 3 = The BP increases, on average, 9.94 mmhg for those who smokes compared to those that does not smoke. This effect is adjusted by all the other variable in the model id effect plot imc effect plot pa id hf effect plot pa imc 150 pa não hf sim DEIO/CEAUL Valeska Andreozzi slide 34 Hypothesis test Analysis of Variance ANOVA partition the total variability in the sample of y i into: (y i y) 2 i }{{} total variability = (ŷ i y) 2 i }{{} variability explained by the regression line + (y i ŷ i) 2 i }{{} variability not explained (residual variation about the fitted line) Variability partitions DEIO/CEAUL Valeska Andreozzi slide 35 13

14 ANOVA (y i y) 2 = (ŷ i y) 2 i i }{{}}{{} Total Regression + i (y i ŷ i ) 2 } {{ } Residual Source Sum of degrees of Mean sum of squares (SS) freedom (df) squares (MS) Regression SSreg = (ŷ i ȳ) 2 m MSregression = SSreg m Residual SSE= (y i ŷ i ) 2 n m 1 MSresidual = SSE n m 1 Total SStotal = (y i ȳ) 2 n 1 MStotal = SStotal n 1 DEIO/CEAUL Valeska Andreozzi slide 36 Hypothesis test ANOVA F = SSreg m SSE n m 1 with n = number of observations and m = number of variables. = MSregression MSresidual F m,n m 1 If there is no linear relationship, the regression sum of square just represent random variation so the regression mean square is another, independent, estimate of σ 2 The F test indicates whether there is evidence of a linear relationship between Y and X F test: Ratio between variability explained by the regression and residual variation This ratio will close to one if there is no an effective relationship and will be larger, otherwise. In the simple linear regression this is equivalent to test H 0 : β 1 = 0 versus H 1 : β 1 0 DEIO/CEAUL Valeska Andreozzi slide 37 14

15 in R Commander ANOVA for multiple regression model Statitics > Fit models > Linear models... Call: lm(formula = pa ~ id + imc + hf, data = multi) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) *** id *** imc hf[t.sim] *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 28 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 3 and 28 DF, p-value: 7.602e-09 DEIO/CEAUL Valeska Andreozzi slide 38 Hypothesis test Wald test Test H 0 : β k = 0 versus H 1 : β k 0 using T statistics T = β k EP( β k ) Under H 0, T follows a t-student distribution with n p degrees of freedom (p = number of model coefficients and n = number of observations) or approximately a normal distribution with zero mean and unity variance. DEIO/CEAUL Valeska Andreozzi slide 39 in R Commander Wald test in R Commander Statitics > Fit models > Linear models... Call: lm(formula = pa ~ id + imc + hf, data = multi) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) *** id *** imc hf[t.sim] *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 28 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 3 and 28 DF, p-value: 7.602e-09 DEIO/CEAUL Valeska Andreozzi slide 40 15

16 Hypothesis test Partial F-test To compare two models that are nested, we can compute partial F-test. Suppose two models M p and M q with, respectively, p and q parameters (p < q). M p and M q are nested models if (M p M q ), i.e, all parameters present in M p are also present in M q. To test H 0 : the subset of variables present in M q that is not present M p are all not significant against H 1 : at least one of the variable in this subset is significant to model Y correspond to test simultaneously that q p parameters are all equal to zero, using partial F-test. F = (SSreg q SSreg p )/(q p) SSE q /(n q) F q p,n q DEIO/CEAUL Valeska Andreozzi slide 41 in R Commander Partial F-test in R Commander Models > Hypothesis test > Compare two models... > anova(linearmodel.3, LinearModel.1) Analysis of Variance Table Model 1: pa ~ id Model 2: pa ~ id + imc + hf Res.Df RSS Df Sum of Sq F Pr(>F) *** --- Signif. codes: 0 *** ** 0.01 * DEIO/CEAUL Valeska Andreozzi slide 42 Confidence Interval 100(1 α)% confidence interval of β s is given by: [ β k t n p,α/2 EP( β k ) ; βk + t n p,α/2 EP( β k )] In R: confint(modelo) In R Commander: Models > Confidence intervals... > Confint(LinearModel.6, level=.95) Estimate 2.5 % 97.5 % (Intercept) id imc hf[t.sim] DEIO/CEAUL Valeska Andreozzi slide 43 16

17 Coefficient of determination (Multiple) Coefficient of determination is a summary measure given by the ratio of the regression sum of squares to the total sum of squares. R 2 = SSreg SStotal R 2 represents the proportion of the total sum of squares explained by the regression. R 2 can be estabilished by the square of the correlation between the y i and the predicted values ŷ i from the model. R 2 lies between 0 and 1. Important note: Do not confuse F-test from ANOVA with R 2. F-test from ANOVA indicates whether there is evidence of linear relationship between Y and X or in other words, that the regression model is significant. R 2 measure the quality of the model for prediction of Y DEIO/CEAUL Valeska Andreozzi slide 44 Coefficient of determination R 2 should not be used as a measure of quality of model fit because its value always increase when a variable is add to the model. For this purpose one can use an adjusted coefficient of determination R 2 a R 2 a = 1 MSresidual MStotal DEIO/CEAUL Valeska Andreozzi slide 45 Variable selection There are several methods to select variable into a model. The most popular are the sequential ones: foward selection, backward deletion and stepwise selection. Forward selection: models are systematically built up by adding variables one by one to the null model compromising just β 0 (intercept) Backward deletion: models are systematically reduced by deleting variables one by one to the full model compromising just β 0 Stepwise selection: it is a combination of the two process mentioned above. For any methods, the most crucial decision is the choice of the stopping-rule. Some choices are the Akaike Information Criteria, which there is no statistical distribution associated to proceed a formal test, or the partial F-test, which the level of significance to add or delete a variable has to be chosen. Let s learn by an example. DEIO/CEAUL Valeska Andreozzi slide 46 17

18 Example Chose package MASS and data set birtwt Data > Data in packages > Read data set from an attached package... Help > Help on active data set (if available)... Recode the variables: change bwt to kg, transform race and smoke to factors Data > Manage variables in active data set > Convert numerical variables to factors... Data > Manage variables in active data set > Compute new variable Fit the model: bwt age + ftv + ht + lwt + ptl + race + smoke + ui Statistics > Fit models > Linear model... Select variables by using a sequential method Models > Stepwise model selection... DEIO/CEAUL Valeska Andreozzi slide 47 Example Select variables by using a foward selection with partial F-test. In each step, add a variable that has the minimum p-value inferior to 0.20 nullmodel <-lm(bwt ~ 1, data=birthwt) add1(nullmodel,scope=~ age +ftv +ht + lwt + ptl + race + smoke + ui, test="f") model1<-lm(bwt ~ ui, data=birthwt) add1(model1,scope=~ age +ftv +ht + lwt + ptl + race + smoke +ui,test="f") model1<-lm(bwt ~ ui+race, data=birthwt) add1(model1,scope=~ age +ftv +ht + lwt + ptl + race + smoke +ui,test="f") model1<-lm(bwt ~ ui+race+smoke, data=birthwt) add1(model1,scope=~ age +ftv +ht + lwt + ptl + race + smoke +ui,test="f") model1<-lm(bwt ~ ui+race+smoke+ht, data=birthwt) add1(model1,scope=~ age +ftv +ht + lwt + ptl + race + smoke +ui,test="f") model1<-lm(bwt ~ ui+race+smoke+ht+lwt, data=birthwt) add1(model1,scope=~ age +ftv +ht + lwt + ptl + race + smoke +ui,test="f") addmodel<-lm(bwt~ ui+race+smoke+ht+lwt,data=birthwt) summary(addmodel) DEIO/CEAUL Valeska Andreozzi slide 48 18

19 Example Select variables by using a backward deletion with partial F-test. In each step, delete a variable that has the maximum p-value superior to 0.25 fullmodel <-lm(bwt ~ age +ftv +ht + lwt + ptl + race + smoke + ui, data=birthwt) drop1(fullmodel,test="f") model2 <-lm(bwt ~ age +ht + lwt + ptl + race + smoke + ui, data=birthwt) drop1(model2,test="f") model2 <-lm(bwt ~ ht + lwt + ptl + race + smoke + ui, data=birthwt) drop1(model2,test="f") model2 <-lm(bwt ~ ht + lwt + race + smoke + ui, data=birthwt) drop1(model2,test="f") dropmodel<-lm(bwt ~ ht + lwt + race + smoke + ui, data=birthwt) summary(dropmodel) DEIO/CEAUL Valeska Andreozzi slide 49 Model check Regression diagnostics are used after fitting to check if a fitted mean function and assumptions are consistent with observed data. The basic statistics here are the residuals or possibly rescaled residuals. If the fitted model does not give a set of residuals that appear to be reasonable, then some aspect of the model, either the assumed mean function or assumptions concerning the variance function, may be called into doubt. DEIO/CEAUL Valeska Andreozzi slide 50 Residuals Using the matrix notation, we begin by deriving the properties of residuals. The basic multiple linear regression model is given by Y = Xβ + ǫ and V ar(ǫ) = σ 2 I X is a known matrix with n rows and p columns, including a column of 1s for the intercept β is the unknown parameter vector p 1 ǫ consists of unobservable errors that we assume are equally variable and uncorrelated DEIO/CEAUL Valeska Andreozzi slide 51 19

20 Residuals We estimate β by ˆβ = (X T X) 1 X T Y and the fitted values Ŷ Ŷ = X ˆβ (1) = X(X T X) 1 X T Y (2) = HY (3) where H is a n n called hat matrix because it transforms the vector of observed responses Y into the vector of fitted responses Ŷ DEIO/CEAUL Valeska Andreozzi slide 52 Residuals The vector of residuals ˆǫ is defined by ˆǫ = Y Ŷ (4) = Y X ˆβ (5) = Y X(X T X) 1 X T Y (6) = (I H)Y (7) DEIO/CEAUL Valeska Andreozzi slide 53 Residuals The errors ǫ are unobservable random variables, assumed to have zero mean and uncorrelated elements, each with common variance σ 2. The residuals ˆǫ are computed quantities that can be graphed or otherwise studied. Their mean and variance, using equation 7, are: E(ˆǫ) = 0 V ar(ˆǫ) = σ 2 (I H) Like the errors, each of the residuals has zero mean, but each residual may have a different variance. Unlike the errors, the residuals are correlated The residuals are linear combinations of the errors. If the errors are normally distributed, so are the residuals. DEIO/CEAUL Valeska Andreozzi slide 54 Residuals In scalar form, the variance of the i th residual is V ar(ˆǫ i ) = ˆσ 2 (1 h ii ) (8) where h ii is the i th diagonal element of H Diagnostic procedures are based on the computed residuals, which we would like to assume behave as the unobservable errors would. DEIO/CEAUL Valeska Andreozzi slide 55 20

21 Residuals All the above story is told to conclude that model validation should be done by standardized residuals Here are some examples DEIO/CEAUL Valeska Andreozzi slide 56 Residuals Here are some examples DEIO/CEAUL Valeska Andreozzi slide 57 Residuals Here are some examples DEIO/CEAUL Valeska Andreozzi slide 58 21

22 Residuals Residual plots: (a) null plot; (b) right-opening megaphone; (c) left-opening megaphone; (d) double outward box; (e) - (f) nonlinearity; (g) - (h) combinations of nonlinearity and nonconstant variance function. DEIO/CEAUL Valeska Andreozzi slide 59 Residuals Working residual Pearson residual Pearson standardized residual r p = r = y i µ i r p = y i µ i σ 2 y i µ i σ2 (1 h ii ) in R rstandard(model, type="pearson") in R Commander Models > Add observations statistics to data The R commander calculates de Studentized residuals (re-normalize the residuals to have unit variance, using a leave-one-out measure of the error variance) DEIO/CEAUL Valeska Andreozzi slide 60 22

23 Plot of residuals Constant variance: plot standardized residuals against their corresponding fitted values (ŷ i ). The points should appear randomly and evenly about zero if assumption is respected. Graphs > Scatterplot... rstudent.linearmodel fitted.linearmodel.2 DEIO/CEAUL Valeska Andreozzi slide 61 Plot of residuals Normality: plot the ranked standardized residuals against inverse normal cumulative distribution values. epartures form normality being indicaed by deviantes of the plot from a straight line. Graphs > Quantile-comparision plot... birthwt$rstudent.linearmodel norm quantiles DEIO/CEAUL Valeska Andreozzi slide 62 23

24 Plot of residuals Independence: plot standardized residuals against the serial order in which the observations were taken. Again, random scatter of points indicates that the assumption is valid. Graphs > Scatterplot... low 0 1 rstudent.linearmodel rstudent.linearmodel obsnumber obsnumber Be carefull with the data set organization... DEIO/CEAUL Valeska Andreozzi slide 63 Plot of residuals The truth is: Data > Manage variables in active data set > Compute new variable birthwt$index <- with(birthwt, sample(1:189)) rstudent.linearmodel index DEIO/CEAUL Valeska Andreozzi slide 64 24

25 Plot of residuals Linearity: plot standardized residuals against individual explanatory variables. Linearity is indicated if all plots exhibit random scatter of equal width about zero. Non-linearity when residuals are plotted against explanatory variables in the model suggest that higher-order terms involving those variables should be added to the model. Systematic patterns exhibited when residuals are plotted against variables that are not included in the model suggest that those variables should be added to the model Graphs > Scatterplot... rstudent.linearmodel rstudent.linearmodel lwt age DEIO/CEAUL Valeska Andreozzi slide 65 Interaction When interaction is present, the association between the risk factor and the outcome variable differs, or depends in some way on the level of a covariate. That is, the covariate modifies the effect of the risk factor. Epidemiologists uses the term effect modifier to describe a variable that interacts with a risk factor. Interaction can be included in a regression model by adding the product term covariate times risk factor. DEIO/CEAUL Valeska Andreozzi slide 66 Interaction Interaction representation Without interaction With interaction group A group B group A group B y (outcome) y (outcome) x (risk factor) x (risk factor) DEIO/CEAUL Valeska Andreozzi slide 67 25

26 in R The cystfibr data frame (package: ISwR) contains lung function data for cystic fibrosis patients (7-23 years old). Call: lm(formula = pemaxlog ~ bmp * sex, data = cystfibr) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-08 *** bmp sex[t.fem] bmp:sex[t.fem] Signif. codes: 0 *** ** 0.01 * Residual standard error: on 21 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 3 and 21 DF, p-value: DEIO/CEAUL Valeska Andreozzi slide 68 26