1 Experimental Statistics - week 10 Chapter 11: Linear Regression and Correlation Note: Homework Due Thursday

2 2-Factor with Repeated Measure -- Model type subject within type time type by time interaction NOTES: type and time are both fixed effects in the current example - we say “subject is nested within type” - Expected Mean Squares given on page 1032

3 The GLM Procedure Dependent Variable: conc Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE conc Mean Source DF Type III SS Mean Square F Value Pr > F type subject(type) <.0001 time <.0001 type*time < Factor Repeated Measures – ANOVA Output

4 2-factor Repeated Measures Source Type III Expected Mean Square type Var(Error) + 5 Var(subject(type)) + Q(type,type*time) subject(type) Var(Error) + 5 Var(subject(type)) time Var(Error) + Q(time,type*time) type*time Var(Error) + Q(type*time) The GLM Procedure Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: conc Source DF Type III SS Mean Square F Value Pr > F * type Error Error: MS(subject(type)) * This test assumes one or more other fixed effects are zero. Source DF Type III SS Mean Square F Value Pr > F subject(type) <.0001 * time <.0001 type*time <.0001 Error: MS(Error)

5 NOTE: Since time x type interaction is significant, and since these are fixed effects we DO NOT test main effects – we compare cell means (using MSE) C T Cell Means

6 The write-up related to the SAS output should be something like the following. Note, that even though we get a significant variance component due to subject(within group) I did not estimate the variance component itself. (I did not give this particular variance component estimation formula.) Note also that since there is a significant interaction between the fixed effects type and time, we do not test the main effects.

7 Dealing with Normality/Equal Variance Issues Normalizing Transformations: - log - square root - Box-Cox transformations Note: the normalizing transformations sometimes also produce variance stabilization

8 Nonparametric “ANOVA” Man-Whitney U – for comparing 2 samples Kruskal-Wallis Test – for comparing >2 samples Friedman’s Test – nonparametric alternative to randomized complete block/ 1-factor repeated measures design

HistogramHistogram displays distribution of 1 variable Scatter Diagram (Scatterplot) Scatter Diagram (Scatterplot) displays joint distribution of 2 variables plots data as “points” in the“x-y plane.”

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11

Association Between Two Variables – indicates that knowing one helps in predicting the other Linear Association – our interest in this course – points “swarm” about a line Correlation Analysis – measures the strength of linear association

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14 (association)

Regression Analysis We want to predict the dependent variable - response variable using the independent variable - explanatory variable - predictor variable DependentVariable(Y) Independent Variable (X) More than one independent variable – Multiple Regression

Correlation Analysis

Correlation Coefficient - measures linear association perfect no perfect negative linear positive relationship relationship relationship

Positive Correlation - - high values of one variable are associated with high values of the other Examples: - father’s height, son’s height - daily grade, final grade r = 0.93 for plot on the left

19 EXAMS I and II

Negative Correlation - - high with low, low with high Examples: - car age, selling price - days absent, final grade r = for plot shown here

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Zero Correlation - - no linear relationship Examples: - height, IQ score r = 0.0 for plot here

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, 0,.5,.99

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26 Calculating the Correlation Coefficient

27 Notation: So --

28 Study Time Exam (hours) Score (X) (Y) The data below are the study times and the test scores on an exam given over the material covered during the two weeks. Find r

29 DATA one; INPUT time score; DATALINES; ; PROC CORR; Var score time; TITLE ‘Study Time by Score'; RUN; PROC PLOT; PLOT time*score; RUN; PROC GPLOT; PLOT time*score; RUN;

30 The CORR Procedure 2 Variables: score time Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum score time Pearson Correlation Coefficients, N = 8 Prob > |r| under H0: Rho=0 score time score time Study Time by Score

31 Plot of score*time. Legend: A = 1 obs, B = 2 obs, etc. score ‚ ‚ 92 ˆ A ‚ 91 ˆ ‚ 90 ˆ ‚ 89 ˆ ‚ 88 ˆ ‚ 87 ˆ ‚ 86 ˆ ‚ 85 ˆ A ‚ 84 ˆ A A ‚ 83 ˆ ‚ 82 ˆ ‚ 81 ˆ A ‚ 80 ˆ A A ‚ 79 ˆ ‚ 78 ˆ ‚ 77 ˆ ‚ 76 ˆ ‚ 75 ˆ ‚ 74 ˆ A ‚ Šƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒƒƒˆƒƒ time

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33 Rejection Region Test Statistic t > t  2 or t <  t  /2 df  n  2 Testing Statistical Significance of Correlation Coefficient

34 Correlation Between Study Time and Score H 0 : There is No Correlation Between Study Time and Score H a : There is a Correlation Between Study Time and Score Rejection Region Test Statistic Conclusion P-value

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36 Correlation measures the strength of the linear relationship between two variables. Correlation requires that both variables be quantitative. r does not change when we change the units of measurement of x, y, or both. Correlation makes no distinction between explanatory and response variables. The correlation coefficient is not resistant to outliers. Properties of Correlations

37 The CORR Procedure Pearson Correlation Coefficients, N = 20 Prob > |r| under H0: Rho=0 math reading math <.0001 reading <.0001 Math vs Reading Scores

38 The CORR Procedure Pearson Correlation Coefficients, N = 20 Prob > |r| under H0: Rho=0 math reading math reading Math vs Reading Scores with Outlier

39 Pearson Correlation Coefficients, N = 14 Prob > |r| under H0: Rho=0 math reading math reading

40 Pearson Correlation Coefficients, N = 14 Prob > |r| under H0: Rho=0 math reading math reading

41 Divorce Rate (per 1000) % in prison on Drug Offenses

IMPORTANT NOTE: Correlation DOES NOT Imply Causation strong association between 2 variables is not enough to justify conclusions about cause and effect best way to get evidence that X causes Y is through a controlled experiment

Regression Analysis

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45 Goal of Regression Analysis: Predict Y from knowledge of X For data such as the Father-Son data, it seems reasonable to assume a model of the form i.e. the conditional means of Y given x follow a straight line

46 Alternative mathematical expression for the “regression model”: In practice, we want to estimate this line from the data.

Which line is “closest” to the points ?

Criterion for measuring “closeness” --- the sum of squared vertical distances from the points to the line Regression (Least Squares) Line --- the line for which this sum-of-squared distance is a minimum

55 Notation Theoretical Model Regression line

56 Data we write

57 NOTE: - this is a calculus problem

58 Least Squares Estimates Computation Formula

59 Study Time Exam (hours) Score (X) (Y) The data below are the study times and the test scores on an exam given over the material covered during the two weeks. Find the equation of the regression line for prediction exam score from study time.

60 The GLM Procedure Dependent Variable: score Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE score Mean Source DF Type I SS Mean Square F Value Pr > F time Source DF Type III SS Mean Square F Value Pr > F time Standard Parameter Estimate Error t Value Pr > |t| Intercept <.0001 time PROC GLM; MODEL score=time; RUN;