3.3. SIMPLE LINEAR REGRESSION: DUMMY VARIABLES 1 Design and Data Analysis in Psychology II Salvador Chacón Moscoso Susana Sanduvete Chaves.

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3.3. SIMPLE LINEAR REGRESSION: DUMMY VARIABLES 1 Design and Data Analysis in Psychology II Salvador Chacón Moscoso Susana Sanduvete Chaves

1. Introduction -Qualitative independent variable. -Quantitative dependent variable. -Two independent groups: simple linear regression. -More than two independent groups: multiple linear regression. -Codification dummy – Fictious variable: -0 = absence; 1 = presence -Example: 1 = woman; 0 = man (absence of being woman). 2

3 1. Introduction Linearity assumption is always accepted. The other assumptions must be tested.

4 2. Parameters estimation Regression equation: Formulas to calculate a and b previously used are still useful. In addition:

5 Example. Those are the results obtained by women (X=0) and men (X=1) in a linguistic ability test. XY Parameters estimation

6 1.Calculate the regression equation in raw scores using the new formulas. 2. Interpret the parameters. 3. Calculate the regression equation in raw scores using the classic formulas. 4. Calculate the regression equation supposing that women were codified with 1 and men with Parameters estimation

7 1. Calculate the regression equation in raw scores using the new formulas. 2. Parameters estimation

8 2. Interpret the parameters. -a is the mean value of Y when X = is the mean result obtained in the linguistic ability test in women group. -b is the change in the mean value of Y in the group X = 1 compared to the group X = 0. The mean result is 3.4 lower in men than in women (concretely, the mean result in men is 4). 2. Parameters estimation

9 3. Calculate the regression equation in raw scores using the classic formulas. XYxyx*yx2x2 y2y ,51,3-0,650,251, ,50,3-0,150,250, ,51,3-0,650,251, ,52,3-1,150,255, ,53,3-1,650,2510, ,5-2,7-1,350,257, ,5-0,7-0,350,250, ,5-1,7-0,850,252, ,5-0,7-0,350,250, ,5-2,7-1,350,257,29 557Σ-8,52,538,1 2. Parameters estimation

10 3. Calculate the regression equation in raw scores using the classic formulas. 2. Parameters estimation

11 4. Calculate the regression equation supposing that women were codified with 1 and men with 0. XY Parameters estimation

12 4. Calculate the regression equation supposing that women were codified with 1 and men with Parameters estimation

13 3. Goodness of fit p = ratio of n 1 related to the whole sample. q = ratio of n 0 n 1 related to the whole sample. p + q = 1

14 3. Goodness of fit Example. Using the same data, calculate the goodness of fit using the two formulas proposed, and interpret the results obtained.

15 3. Goodness of fit Ratio of total variability of Y explained by X. 75.8% of variability of linguistic ability is explained by sex.

16 4. Model validation -  Null hypothesis is rejected. The variables are related. The model is valid. –  Null hypothesis is accepted. The variables are not related. The model is not valid. (k = number of independent variables)

17 4. Model validation Example. Using the same data, conclude about the model validation.

18 4. Model validation

19 4. Model validation 19  Conclusión: Null hypothesis is rejected. The variables are related. The model is valid. There is a statistically significant relationship between sex and linguistic ability.

4. Model validation Which is the final conclusion? Significant effect Non-significant effect High effect size ( ≥ 0.67) The effect probably exists The non- significance can be due to low statistical power Low effect size ( ≤ 0.18) The statistical significance can be due to an excessive high statistical power The effect probably does not exist 20

4. Model validation Which is the final conclusion? Significant effect Non-significant effect High effect size R 2 = ( ≥ 0.67) The effect probably exists The non- significance can be due to low statistical power Low effect size ( ≤ 0.18) The statistical significance can be due to an excessive high statistical power The effect probably does not exist 21

22 5. Significance Of Pearson correlation coefficient:

23 5. Significance Of the slope:

24 5. Significance Of samples contrast:

25 5. Significance –  Null hypothesis is rejected. The model is valid. The slope is statistically different from 0. There is, therefore, relationship between variables. –  Null hypothesis is accepted. The model is not valid. The slope is statistically equal to 0. There is not, therefore, relationship between variables. 25

26 5. Significance Example. Calculate t using the different formulas of t and conclude about the significance.

27 5. Significance

28 5. Significance XYxyx*yx2x2 y2y2 Y2Y2 X2X ,51,3-0,650,251, ,50,3-0,150,250, ,51,3-0,650,251, ,52,3-1,150,255, ,53,3-1,650,2510, ,5-2,7-1,350,257, ,5-0,7-0,350,250, ,5-1,7-0,850,252, ,5-0,7-0,350,250, ,5-2,7-1,350,257, ∑-8,52,538,13635

29 5. Significance

30 5. Significance 30 Conclussion: Null hypothesis is rejected. The model is valid. There is statistically significant relationship between sex and linguistic ability.