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❘❙❚■ Chapter 24 | Regression Analysisbinary, or categorical). Again, we use an i subscript to denote the value takenby the ith individual on each variable. The multiple linear regression model canbe written:y i= a + b 1x 1i+ b 2x 2i+ ... + b px pi+ e i(1)where e iis called a residual, and represents variables other than the p predictorsthat affect y. Various assumptions are made about the residuals, namely that theyfollow a normal distribution and have constant variance across different values ofthe predictor variables (the assumption of homoskedasticity). The adequacy of theseassumptions should be checked by examining plots of the estimated residuals.a is called the intercept and is interpreted as the mean of y when all the xsequal 0. Of more interest are the bs, referred to as the regression coefficients;b k(k = 1, 2, …, p). Regression coefficients are interpreted as either thepredicted change in y for a one unit increase in x kif it is a continuous variable,or the difference between two groups if it is a binary variable, adjusting for theeffects of the other predictor variables in the model. P-values and confidenceintervals (CIs) can be obtained for each regression coefficient to assess whetherthe associated predictors have statistically significant effects on the response.ExampleTo illustrate the use of multiple linear regression, let us consider a randomizedcontrolled trial on 220 depressed residents aged ≥65 without severe cognitiveimpairment, conducted by Llewellyn-Jones et al [1]. The primary endpoint(response variable) was the geriatric depression scale score at follow-up: a higherdepression score meant more depression features. They used multiple linearregression analysis to evaluate the effect of intervention on the depression scalescore at follow-up, while controlling for the other independent variablesmeasured. Table 1 presents the estimates of the regression coefficients, their95% CIs, and associated P-values for the intervention variable and somesignificant predictors from this study. Multiple linear regression analysis founda significant intervention effect after controlling for possible confounders, withthe intervention group showing an average improvement of 1.87 points on thegeriatric depression scale compared with the control group (95% CI 0.76, 2.97;P = 0.0011). The regression coefficients also tell us that geriatric depression scalescore at follow-up increases by 0.73 points for every geriatric depression scalescore at baseline, by 0.55 points for neuroticism score, and 0.10 points for everyyear of age, but decreases by 0.54 points for every score of basic functional abilityat baseline.276
<strong>Clinical</strong> <strong>Trials</strong>: A Practical Guide ■❚❙❘Table 1. Factors affecting geriatric depression scale score at follow-up, determined by multiple linearregression analysis: extracted results from [1].Variable Regression coefficient (95% CI) P-valueIntervention group vs control –1.87 (–2.97, –0.76) 0.0011Baseline geriatric depression scale score 0.73 (0.56, 0.91)
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❘❙❚■ ContributorsStephen L
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Clinical Trials: A Practical Guide
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■■❚❙❘Chapter 1RandomizedC
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■■❚❙❘Chapter 4EndpointsAm
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■■❚❙❘Part IIAlternativeTr
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❘❙❚■ Chapter 16 | Multicent
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■■❚❙❘Part IIIBasics ofSta
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■■❚❙❘Chapter 17Types of D
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■■❚❙❘Chapter 19Comparison
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■■❚❙❘Chapter 20Comparison
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■■❚❙❘Chapter 33Trial Prof
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■■❚❙❘Chapter 35Use of Tab
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■■❚❙❘GlossaryGlossary453
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■■❚❙❘IndexIndex467
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