15.7Regarding regression coefficientsBefore moving on to discuss the assumptions underlying linear regression and what you can do to check ifthey’re being met, there’s two more topics I want to briefly discuss, both of which relate to the regressioncoefficients. The first thing to talk about is calculating confidence intervals for the coefficients; after that,I’ll discuss the somewhat murky question of how to determine which of predictor is most important.15.7.1Confidence intervals for the coefficientsLike any population parameter, the regression coefficientsbcannot be estimated with complete precisionfrom a sample of data; that’s part of why we need hypothesis tests. Given this, it’s quite useful to be ableto report confidence intervals that capture our uncertainty about the true value ofb. This is especiallyuseful when the research question focuses heavily on an attempt to find outhowstrongly variableXis related to variableY, since in those situations the interest is primarily in the regression weightb.6You can change the kind of correction it applies by specifying thep.adjust.methodargument.-472-

Fortunately, confidence intervals for the regression weights can be constructed in the usual fashion,CIpbq “ˆb˘´tcritˆsepˆbq¯wheresepˆbqis the standard error of the regression coefficient, andtcritis the relevant critical value of theappropriatetdistribution. For instance, if it’s a 95% confidence interval that we want, then the criticalvalue is the 97.5th quantile of atdistribution withN´K´1 degrees of freedom. In other words, this isbasically the same approach to calculating confidence intervals that we’ve used throughout. To do thisinRwe can use theconfint()function. There arguments to this function are•object. The regression model (lmobject) for which confidence intervals are required.•parm. A vector indicating which coefficients we should calculate intervals for. This can be either avector of numbers or (more usefully) a character vector containing variable names. By default, allcoefficients are included, so usually you don’t bother specifying this argument.•level. A number indicating the confidence level that should be used. As is usually the case, thedefault value is 0.95, so you wouldn’t usually need to specify this argument.So, suppose I want 99% confidence intervals for the coefficients in theregression.2model. I could dothis using the following command:>confint( object = regression.2,+level = .99+)0.5 %99.5 %(Intercept) 117.9755724 133.9555593dan.sleep-10.4044419-7.4960575baby.sleep-0.70168680.7227357Simple enough.15.7.2Calculating standardised regression coefficientsOne more thing that you might want to do is to calculate “standardised” regression coefficients,often denotedβ.The rationale behind standardised coefficients goes like this.In a lot of situations,your variables are on fundamentally different scales. Suppose, for example, my regression model aims topredict people’s IQ scores, using their educational attainment (number of years of education) and theirincome as predictors.Obviously, educational attainment and income are not on the same scales: thenumber of years of schooling can only vary by 10s of years, whereas income would vary by 10,000s ofdollars (or more).

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Term

Fall

Professor

AgnieszkaKopinska

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