1. From last time….

2. Basic Biostats Topics Summary Statistics –mean, median, mode –standard deviation, standard error Confidence Intervals Hypothesis Tests –t-test (paired and unpaired) –Chi-Square test –Fisher’s exact test

3. More Advanced Linear Regression Logistic Regression Repeated Measures Analysis Survival Analysis Analyzing fMRI data

4. General Biostatistics References Practical Statistics for Medical Research. Altman. Chapman and Hall, 1991. Medical Statistics: A Common Sense Approach. Campbell and Machin. Wiley, 1993 Principles of Biostatistics. Pagano and Gauvreau. Duxbury Press, 1993. Fundamentals of Biostatistics. Rosner. Duxbury Press, 1993.

5. Lecture 3: Linear Regression Elizabeth Garrett esg@jhu.edu Child Psychiatry Research Methods Lecture Series

6. Introduction Simple linear regression is most useful for looking at associations between continuous variables. We can evaluate if two variables are associated linearly. We can evaluate how well we can predict one of the variables if we know the other.

7. Motivating Example (Tierney et al. 2001) Is there an association between total sterol level and ADI scores in autistic children? Hypothesis: Children with lower sterol levels will tend to have poorer performance (i.e. higher scores) on the following components of the ADI: –social –nonverbal –repetitive

8. Preliminary Data 9 individuals with autism Some have been on cholesterol supplementation (7 out of 9) Mean age: 14 Age range: 8 - 32 years Sterol is a continuous variable ADI scores are continuous variables

9. Statistical Language Need to choose what variable is the predicted (Y) and which is the predictor (X). Y: outcome, dependent variable, endogenous variable X: covariate, predictor, regressor, explanatory variable, exogenous variable, independent variable. Our example?

10. How can we conclude if there is or is not an association between sterol and the ADI scores?

11. One approach: Correlation Correlation is a measure of LINEAR association between two variables. It takes values from -1 to 1. Often notated r or  r = 1  perfect positive correlation r = -1  perfect negative correlation r = 0  no correlation

12. r = 0.95 r = 0.09 r = 0.77 r = -0.95

13. Correlation between ADI measures and Sterol r = -0.70 r = 0.06 r = -0.85

14. Related to r: R 2 R 2 = % of variation in Y explained by X. Example: –Correlation between nonverbal score and sterol is -0.85. –R 2 is 0.85 2 = 0.73 –73% of the variation in nonverbal score is explained by sterol Gives a sense of the value of sterol in predicting nonverbal score Other examples –R 2 between sterol and social is 0.49 –R 2 between sterol and repetitive is 0.004

15. Simple Linear Regression (SLR) Approach (1) Fits “best” line to describe the association between Y and X (note: straight line) (2) Line can be described by two numbers - intercept - slope (3) By-product of regression: correlation measures how close points fall from the line. (4) Why “simple”? Only one X variable.

16. Intercept = 24.8 Slope = -0.01

17. SLR answers two questions…. Association? –Does nonverbal score tend to decrease on average when sterol increases? –Is slope different than zero? Prediction? –Can we predict nonverbal score if we know sterol level? –Is the correlation (or R 2 ) high? You CAN have association with low correlation!

18. Equation of a line:  0 : Intercept –  0 is the estimated nonverbal score if it were possible to have a sterol level of 0 (nonsensical in this case). –  0 calibrates height of line  1 : Slope –  1 is the estimated change in nonverbal score for a one unit change in sterol –  1 the estimated difference in nonverbal score comparing two kids whose sterol levels differ by one. –We usually use  1 as our measure of association

19. The slope,  1 Is  1 different than zero? Are each of these reasonable given the data that we have observed?

20. Evaluating Association  1 is a “statistic,” similar to a sample mean, and as such has a precision estimate. The precision estimate is called the standard error of  1. Denoted se(  1 ). We look at how large  1 is compared to its standard error  1 is often called a “regression coefficient” or a “slope.”

21. General Rule If, then we say that  1 is statistically significantly different than zero. T-test interpretation: H 0 :  1 = 0 H a :  1  0 If is true, then p-value less than 0.05. Intuition:  1 is large compared to its precision  not likely that  1 is 0.

22. For large samples….

23. ADI Nonverbal and Sterol ------------------------------------------------------------------------------ nonvrb | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- totster | -.0099066.0022804 -4.344 0.003 -.0152988 -.0045144 _cons | 24.84349 2.578369 9.635 0.000 18.74661 30.94036 ------------------------------------------------------------------------------ 11 00 se(  1 ) pvalue R-squared = 0.73 Outcome Predictor

24. Interpretation “Comparing two autistic kids whose sterol levels differ by 1, we estimate that the one with lower sterol will have an ADI nonverbal score that is higher by 0.01 points.” Put it in “real” units: “Comparing two autistic kids whose sterol levels differ by 200, we estimate that the child with the lower sterol level will have an ADI nonverbal score that is higher by 2 points.” (Note: 200 x 0.01 = 2.0)

25. A few other details... 95% Confidence interval interpretation:  1  2se(  1 ) does not include zero.  1 /se(  1 ) is called the –“t-statistic” –“Z-statistic” If you have small sample (i.e. fewer than 50 individuals), need to use a “t-correction.”

26. Relationship between correlation and SLR Testing that correlation is equal to zero is equivalent to testing that the slope is equal to zero. Can have strong association and low correlation r = 0.93  1 = 1.86 pvalue < 0.001 r = 0.55  1 = 1.88 pvalue < 0.001

27. Additional Points (1) Association measured is LINEAR r = 0.02

28. Additional Points (2) Difference (i.e. distance) between observed data and fitted line is called a residual, . 1. 0.74 2. -0.95 3. -2.53 4. 3.01 5. 2.52 6. 0.45 7. -3.15 8. -0.07 9. 0.59. 33 55

29. Additional Points (3) Often see model equation as Generically, Refers to regression line Refer to observed data

30. Additional Points (4) Spread of points around line is assumed to be constant (i.e. variance of residuals is constant) BAD!

31. Multiple Linear Regression More than one X variable Generally the same, except –Can’t make plots in multi-dimensions –Interpretation of  ’s is somewhat different

32. Other ADI and Sterol SLRs How is age when supplementation began related to sterol? How is age when supplementation began related to nonverbal score?

34. How might this change our previous result? Sterol Nonverbal Score What if age when cholesterol supplementation began is associated with both sterol level and nonverbal score? Is it correct to conclude that total sterol level is associated with nonverbal score? Supplementation Age

35. We can “adjust”! ------------------------------------------------------------------------------ nonvrb | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- sterol | -.0105816.0022118 -4.784 0.003 -.0159937 -.0051696 agester |.1570626.1158509 1.356 0.224 -.1264143.4405394 _cons | 23.81569 2.551853 9.333 0.000 17.57153 30.05985 ------------------------------------------------------------------------------

36. Interpretation of Betas Now that we have “adjusted” for age at supplementation, we need to include that in our result: “Comparing two kids who began cholesterol supplementation at the same age and whose sterol levels differ by 250 units, we estimate that the child with the lower sterol level will have an ADI nonverbal score higher by 2 points.” “Adjusting for age at supplementation, comparing two kids whose sterol levels differ by 250 units, we estimate…” “Controlling for age at supplementation …..” “Holding age at supplementation constant…..”

37. Collinearity If two variables are –correlated with each other –correlated with the outcome Then, when combined in a MLR model, it could happen that –neither is significant –only one is significant –both remain significant

38. ADI and Sterol We say that cholesterol time and sterol are “collinear.” Correlation Matrix | nonvrb sterol agester ---------+--------------------------- nonvrb | 1.0000 sterol | -0.8541 1.0000 agester | 0.0531 0.2251 1.0000

39. Summing up example…. After adjusting for age at supplementation, it appears that sterol is still a significant predictor of ADI nonverbal score. BUT! –Only NINE observations! With more, we would almost CERTAINLY see even stronger associations! –We haven’t controlled for other potential confounders: length of time on supplementation nonverbal score prior to supplementation