Ct lecture 12. simple linear regression analysis
0 likes78 views
This document summarizes a workshop on simple linear regression analysis. It discusses using linear regression to model the relationship between femoral neck bone density (BMD) and age or weight in women. Examples are analyzed using R to estimate regression parameters, interpret results, and check assumptions. Linear regression can be used to understand risk factor effects, predict outcomes, and its assumptions like linearity and normal errors should always be checked.
Ct lecture 12. simple linear regression analysis
- 1. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Simple linear regression analysis Tuan V. Nguyen Professor and NHMRC Senior Research Fellow Garvan Institute of Medical Research University of New South Wales Sydney, Australia
- 2. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 What we are going to learn … • Examples • Purposes of linear regression analysis • Questions of interest • Model parameters • R analysis • Interpretation
- 3. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Femoral neck bone density and age women = subset(vd, sex==2) plot(fnbmd ~ age, pch=16) abline(lm(fnbmd ~ age)) 20 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 age fnbmd
- 4. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Weight and femoral neck bone density plot(fnbmd ~ weight, pch=16) abline(lm(fnbmd ~ weight)) 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 weight fnbmd
- 5. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Correlation analysis • Assessment of a relationship • The coefficient of correlation: a measure of the relationship • We want to know more … – The magnitude of effect of a predictor variable on the outcome – Prediction of outcome by using the predictor variable(s)
- 6. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Our interests • Finding a statistical model that decribes the relationship between age, weight, and BMD • Adjustment of effect • Prediction Linear regression model
- 7. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Weight and femoral neck bone density 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 weight fnbmd
- 8. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 We can also describe the line in terms of a slope and an intercept • The slope the change in the y-value for a unit change in the x-value. In this simple situation we can think of this as the change in the height of the line as we progress along the x- axis • The intercept is the height of the line when x = 0 (x1, y1) (x2, y2) x-axis y-axis 2 1 2 1 y y y slope x x x Δ − = = Δ − 0
- 9. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Linear regression: model • Y : random variable representing a response • X : random variable representing a predictor variable (predictor, risk factor) – Both Y and X can be a categorical variable (e.g., yes / no) or a continuous variable (e.g., age). – If Y is categorical, the model is a logistic regression model; if Y is continuous, a simple linear regression model. • Model Y = a + bX + e a : intercept b : slope / gradient e : random error (variation between subjects in y even if x is constant, e.g., variation in cholesterol for patients of the same age.)
- 10. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 • The relationship is linear in terms of the parameter; • X is measured without error; • The values of Y are independently from each other (e.g., Y1 is not correlated with Y2) ; • The random error term (e) is normally distributed with mean 0 and constant variance. Linear regression: assumptions
- 11. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Criteria of estimation Y X i i bx a y + = ˆ i i i y y d ˆ − = yi The goal of least square estimator (LSE) is to find a and b such that the sum of d2 is minimal.
- 12. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 • We could try fitting a line “by eye” • But everyone’s best guess would probably be different • We want consistency 0 5 10 15 20 25 0 20 40 60 80 100 x y
- 13. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Estimating parameters by R • Our interest: relationship between BMD and weight • Model: BMD = a + b*weight + e • We want to estimate a and b • R language lm(bmd ~ weight)
- 14. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 R analysis > m1 = lm(fnbmd ~ weight) > summary(m1) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.4699822 0.0310144 15.15 < 2e-16 *** weight 0.0049416 0.0006041 8.18 1.95e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.1152 on 556 degrees of freedom Multiple R-squared: 0.1074, Adjusted R-squared: 0.1058 F-statistic: 66.9 on 1 and 556 DF, p-value: 1.945e-15
- 15. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Interpretation of outputs Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.4699822 0.0310144 15.15 < 2e-16 *** weight 0.0049416 0.0006041 8.18 1.95e-15 *** • Remember our model: BMD = a + b*weight • Our equation: BMD = 0.47 + 0.0049*weight • Interpretation: 1 kg increase in weight was associated with a 0.0049 g/cm2 increase in BMD. The association is statistically significant (P < 0.0001)
- 16. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 BMD = 0.47 + 0.0049*weight 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Weight FNBMD
- 17. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Analysis of variance • BMD = a + b*weight + e • Observed variation = model + random “Variation” = sum of squares • SST = total sum of squares SSR = sum of squares due to the regresson model SSE = sum of squares due to random component • SST = SSR + SSE • R2 = SSR / SST
- 18. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Partitioning of variations: geometry BMD Weight mean SSR SSE SST SST = SSR + SSE
- 19. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Partitioning of variation by R • Total SS = 0.8883 + 7.3819 = 8.2702 • R2 = 0.8883 / 8.2702 = 0.107 > m1 = lm(fnbmd ~ weight) > anova(m1) Analysis of Variance Table Response: fnbmd Df Sum Sq Mean Sq F value Pr(>F) weight 1 0.8883 0.88829 66.905 1.945e-15 *** Residuals 556 7.3819 0.01328 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
- 20. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Interpretation of outputs Residual standard error: 0.1152 on 556 degrees of freedom Multiple R-squared: 0.1074, Adjusted R-squared: 0.1058 F-statistic: 66.9 on 1 and 556 DF, p-value: 1.945e-15 • R2 = 0.107 • Interpretation: Approximately 11% of BMD variance could be accounted for by body weight
- 21. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Variance of BMD after adjusting for weight • Mean square (MS) = sum of squares / (degrees of freedom) • MS(residuals) = 7.3819 / 556 = 0.01328 Þ Variance of BMD after adjusting for weight is 0.01328 (variance of BMD before the adjustment: 0.01485 > m1 = lm(fnbmd ~ weight) > anove(m1) Analysis of Variance Table Response: fnbmd Df Sum Sq Mean Sq F value Pr(>F) weight 1 0.8883 0.88829 66.905 1.945e-15 *** Residuals 556 7.3819 0.01328 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
- 22. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Prediction of BMD by weight • The model: BMD = 0.47 + 0.0049*weight • Without the knowledge of weight, the mean BMD is 0.72 g/cm2 • With knowledge of weight, we know that BMD is dependent on weight • Weight = 50 kg, BMD = 0.47 + 0.0049*50 = 0.72 g/cm2 Weight = 40 kg, BMD = 0.47 + 0.0049*40 = 0.67 g/cm2 Weight = 60 kg, BMD = 0.47 + 0.0049*60 = 0.76 g/cm2
- 23. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Checking model assumptions par(mfrow=c(2,2)) plot(m1) 0.65 0.75 0.85 -0.4 -0.2 0.0 0.2 0.4 Fitted values Residuals Residuals vs Fitted 390 3 141 -3 -2 -1 0 1 2 3 -3 -1 0 1 2 3 Theoretical Quantiles Standardized residuals Normal Q-Q 390 3 141 0.65 0.75 0.85 0.0 0.5 1.0 1.5 Fitted values Standardized residuals Scale-Location 390 3141 0.000 0.010 0.020 0.030 -3 -1 1 2 3 Leverage Standardized residuals Cook's distance Residuals vs Leverage 40 27 13
- 24. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Be careful! Anscrombe’s data Frank Anscombe devised 4 sets of X-Y pairs
- 25. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Mean and SD of Anscombe’s data
- 26. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Correlation between X and Y: Anscombe’s data
- 27. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Regression analysis: Anscombe’s data
- 28. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 But …
- 29. Workshop on Analysis of Clinical Studies – Can Tho University of Medicine and Pharmacy – April 2012 Summary • Simple linear regression model is used for – Understanding the effect of a risk factor or determinant on an outcome variable – Predicting an outcome variable • It’s appropriate when the functional relationship is linear • Always check assumptions!