1. Reasoning in Psychology Using Statistics
2019

2. Don’t forget quiz 8 due this Friday
Annoucements

3. Measures of Error in Regression
The linear equation isn’t the whole thing
Also need a measure of error
Y = X(.5) + (2.0) + error Y X 1 2 3 4 5 6
Y = X(.5) + (2.0)
+ error
Same line, but different relationships (strength difference) Y X 1 2 3 4 5 6
Measures of Error in Regression

4. Measures of Error in Regression
The linear equation isn’t the whole thing
Also need a measure of error
Three common measures of error
r2 (r-squared)
R-squared (r2) represents the percent variance in Y accounted for by X
Sum of the squared residuals = SSresidual= SSerror
Compute the difference between the predicted values and the observed values (“residuals”)
Square the differences
Add up the squared differences
Standard error of estimate Y X 1 2 3 4 5 6
Measures of Error in Regression
Statistics by Jim: Standard Error of the Regression vs. R-squared

5. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror Y X 1 2 3 4 5 6
6.2
1.6
5.3
3.45
These are all points on the prediction line
X Y
6.2
= (0.92)(6)+0.688
1.6
= (0.92)(1)+0.688
5.3
= (0.92)(5)+0.688
3.45
= (0.92)(3)+0.688
3.45
= (0.92)(3)+0.688
mean
3.6
4.0
Measures of Error in Regression

6. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
residuals
X Y
These are deviations between the points on the prediction line and the actual observed values (in the Y direction) Y X 1 2 3 4 5 6
6.2 =
-0.20
1.6 =
0.40
5.3 =
0.70
3.45 =
0.55
3.45
-1.45 =
mean
3.6
4.0
Quick check
0.00
Measures of Error in Regression

7. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
X Y
6.2
-0.20
0.04
1.6
0.40
0.16
5.3
0.70
0.49
3.45
0.55
0.30
3.45
-1.45
2.10
mean
3.6
4.0
0.00
3.09
SSERROR
Measures of Error in Regression

8. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
4.0
0.0
16.0
= SSY
X Y
6.2
-0.20
0.04
1.6
0.40
0.16
5.3
0.70
0.49
3.45
0.55
0.30
3.45
-1.45
2.10
mean
3.6
4.0
0.00
3.09
SSERROR
Don’t get these confused with each other
Measures of Error in Regression

9. Measures of Error in Regression
Standard error of the estimate represents the average deviation from the line
Can think of it as the standard deviation of the residuals Y X 1 2 3 4 5 6
df = n - 2
Measures of Error in Regression

10. Measures of Error in Regression
SPSS Regression output gives you a lot of stuff r2
percent variance in Y accounted for by X
Standard error of the estimate
the average deviation from the line
SSresiduals or SSerror
Measures of Error in Regression

11. Chi-Square Test for Independence
A manufacturer of watches takes a sample of 200 people. Each person is
classified by age and watch type preference (digital vs. analog).
Young (under 30)
Old (over 30)
The question: Is there a relationship between age and watch preference?
Chi-Square Test for Independence

12. Chi-Square Test for Independence
A manufacturer of watches takes a sample of 200 people. Each person is
classified by age and watch type preference (digital vs. analog).
Table: “Crosstabs”
Young (under 30)
Old (over 30)
The question: Is there a relationship between age and watch preference?
Chi-Square Test for Independence

13. Decision tree Chi-square test of independence (χ2 lower-case chi )
Describing the relationship between two categorical variables or
Young
Old or
Decision tree

14. Chi-Squared Test for Independence
A manufacturer of watches takes a sample of 200 people. Each person is
classified by age and watch type preference (digital vs. analog). The
question: is there a relationship between age and watch preference?
Chi-Squared Test for Independence

15. Chi-Squared Test for Independence
A manufacturer of watches takes a sample of 200 people. Each person is
classified by age and watch type preference (digital vs. analog). The
question: is there a relationship between age and watch preference?
Step 1: State the hypotheses and select an alpha level
H0: Preference is independent of age (“no relationship”)
HA: Preference is related to age (“there is a relationship”)
We’ll set α = 0.05
Observed scores
Chi-Squared Test for Independence

16. Chi-Squared Test for Independence
Step 2: Compute your degrees of freedom & get critical value
df = (#Columns - 1) * (#Rows - 1)
= (3-1) * (2-1) = 2
Go to Chi-square statistic table and find the critical value
The critical chi-squared value is 5.99
For this example, with df = 2, and α = 0.05
Chi-Squared Test for Independence

17. Chi-Squared Test for Independence
As df gets larger, need larger X2 value for significance. Number of cells get larger.
X2 α = .05
5.99
7.81
11.07
14.07
Chi-Squared Test for Independence

18. Chi-Squared Test for Independence
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies
Observed scores
Chi-Squared Test for Independence

19. Chi-Squared Test for Independence
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies
Observed scores
Spot check: make sure the row totals and column totals add up to the same thing
Chi-Squared Test for Independence

20. Chi-Squared Test for Independence
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies (in each cell)
Observed scores
Expected scores 70 56 14 30 24 6
Under 30
Over 30
Digital
Analog
Undecided
Chi-Squared Test for Independence

21. Chi-Squared Test for Independence
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies (in each cell)
Observed scores
Expected scores 70 56 14
“expected frequencies” - if the null hypothesis is correct, then these are the frequencies that you would expect 30 24 6
Under 30
Over 30
Digital
Analog
Undecided
Chi-Squared Test for Independence

22. Chi-Squared Test for Independence
Step 4: compute the χ2
Find the residuals (fo - fe) for each cell
Chi-Squared Test for Independence

23. Computing the Chi-square
Step 4: compute the χ2
Find the residuals (fo - fe) for each cell
Computing the Chi-square

24. Computing the Chi-square
Step 4: compute the χ2
Find the residuals (fo - fe) for each cell
Square these differences
Computing the Chi-square

25. Computing the Chi-square
Step 4: compute the χ2
Find the residuals (fo - fe) for each cell
Square these differences
Divide the squared differences by fe
Computing the Chi-square

26. Computing the Chi-square
Step 4: compute the χ2
Find the residuals (fo - fe) for each cell
Square these differences
Divide the squared differences by fe
Sum the results
Computing the Chi-square

27. Chi-Squared, the final step
A manufacturer of watches takes a sample of 200 people. Each person is
classified by age and watch type preference (digital vs. analog). The
question: is there a relationship between age and watch preference?
Step 5: Compare this computed statistic (38.09) against the critical value (5.99) and make a decision about your hypotheses
here we reject the H0 and conclude that there is a relationship between age and watch preference
Chi-Squared, the final step

28. Chi square as a statistical test
Our “generic test statistic”
Our chi-square test statistic
each cell = observed difference
difference expected by chance
Chi square as a statistical test

29. Chi-Square Test in SPSS
Analyze  Descriptives  Crosstabs
Chi-Square Test in SPSS

30. Chi-Square Test in SPSS
Analyze  Descriptives  Crosstabs
Click this to get the expected frequencies and residuals
Click this to get bar chart of the results
Chi-Square Test in SPSS

31. Chi-Square Test in SPSS

32. In lab: Gain experience using and interpreting Chi-square procedures
Questions?
Chi-squared test: (~12 mins)
Chi-squared test: (~38 mins)
Chi-squared in SPSS:
Chi-squared distribution:
Wrap up