# 1 STA 536 – Experiments with a Single Factor Regression and ANOVA.

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1 STA 536 – Experiments with a Single Factor Regression and ANOVA

2 STA 536 – Experiments with a Single Factor Relation between x and y Correlation r : is there a relationship between 2 variables? Regression: how well a certain independent variable predict dependent variable? CORRELATION CAUSATION In order to infer causality: manipulate independent variable and observe effect on dependent variable 2

3 STA 536 – Experiments with a Single Factor In the real world… r is never 1 or –1 find best fit of a line in a cloud of observations: Principle of least squares ε y = ax + b ε = residual error =, true value =, predicted value 3

4 STA 536 – Experiments with a Single Factor The relationship between x and y : Finding a and b Population: Model: Solution least squares minimisation: 4

5 STA 536 – Experiments with a Single Factor The relationship between x and y 5

6 STA 536 – Experiments with a Single Factor What can the model explain? Total variance = predicted variance + error variance ) ˆ ( 2 ˆ 2 ii yy yy sss 2 6

7 STA 536 – Experiments with a Single Factor

8 An example: Number of menstrual cycles per year (CYC) in 3 groups of women Controls Joggers Runners Controls Joggers Runners 1 13.50 13.50 13.00 14 12.00 10.00 9.50 2 13.50 13.00 12.50 15 11.50 10.00 9.50 3 13.00 12.50 12.00 16 11.50 10.00 9.00 4 13.00 12.00 11.50 17 11.00 10.00 9.00 5 13.00 12.00 11.00 18 11.00 9.50 9.00 6 12.50 12.00 11.00 19 10.50 9.50 8.00 7 12.50 11.50 10.50 20 10.50 9.00 8.00 8 12.50 11.00 10.50 21 10.50 8.50 7.00 9 12.50 11.00 10.50 22 10.00 8.00 6.00 10 12.00 11.00 10.00 23 10.00 7.50 6.00 11 12.00 11.00 10.00 24 10.00 6.50 5.00 12 12.00 10.50 10.00 25 9.50 6.00 5.00 13 12.00 10.50 10.00 26 9.00 5.00 4.00 8

9 STA 536 – Experiments with a Single Factor Descriptive Statistics Obs Mean Std Dev Minimum Maximum ------------------------------------------------------------------------------------- Controls 26 11.5769231 1.2703724 9.0000000 13.5000000 Joggers 26 10.0576923 2.1416205 5.0000000 13.5000000 Runners 26 9.1346154 2.3939828 4.0000000 13.0000000 ------------------------------------------------------------------------------------- Mean 10.256 Std Dev 2.2138 9

10 STA 536 – Experiments with a Single Factor Boxplot (Minitab – Graph – Boxplot) 10

11 STA 536 – Experiments with a Single Factor Data, grand mean, group means 11

12 STA 536 – Experiments with a Single Factor 12

13 STA 536 – Experiments with a Single Factor 13

14 STA 536 – Experiments with a Single Factor 14

15 STA 536 – Experiments with a Single Factor SS total =SS group +SS error 15

16 STA 536 – Experiments with a Single Factor

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