1. Chi Square & Correlation

2. Nonparametric Test of Chi2
Used when too many assumptions are violated in T-Tests:
Sample size too small to reflect population
Data are not continuous and thus not appropriate for parametric tests based on normal distributions.
χ2 is another way of showing that some pattern in data is not created randomly by chance.
X2 can be one or two dimensional.
X2 deals with the question of whether what we observed is different from what is expected

3. Calculating X2
What would a contingency table look like if no relationship exists between gender and voting for Bush? (i.e. statistical independence)
Male
Female
Voted for Bush 25 50
Voted for Kerry 50 50 50
100
NOTE: INDEPENDENT VARIABLES ON COLUMS AND DEPENDENT ON ROWS

4. Calculating X2
What would a contingency table look like if a perfect relationship exists between gender and voting for Bush?
Male
Female
Voted for Bush 50
Voted for Kerry

5. Calculating the expected value
The expected frequency of the cell in the ith row and jth column
Fi = The total in the ith row marginal
Fj = The total in the jth column marginal
N = The grand total, or sample size for the entire table
Expected Voted for Bush = 50x50 / 100 = 25

6. Nonparametric Test of Chi2
Again, the basic question is what you are observing in some given data created by chance or through some systematic process?
O= Observed frequency
E= Expected frequency

7. Nonparametric Test of Chi2
The null hypothesis we are testing here is that the proportion of occurrences in each category are equal to each other (Ho: B=K). Our research hypothesis is that they are not equal (Ha: B =K).
Given the sample size, how many cases could we expect in each category (n/#categories)? The obtained/critical value estimation will provide a coefficient and a Pr. that the results are random.

8. Let’s do a X2 50 (50-25)2/25=25 (0 - 25)2 /25=25 (50-25)2 /25=25
Male
Female
Voted for
Bush 50
Voted
For Kerry
What would X2 be when there is statistical independence?

9. Let’s corroborate with SPSS

10. Testing for significance
How do we know if the relationship is statistically significant?
We need to know the df (df= (R-1) (C-1) )
(2-1)(2-1)= 1
We go to the X2 distribution to look for the critical value (CV= 3.84)
We conclude that the relationship gender and voting is statistically significant.
Male
Female
Voted for
Bush 20 30
Voted for
Kerry
X2= 4

11. When is X2 appropriate to use?
X2 is perhaps the most widely used statistical technique to analyze nominal and ordinal data
Nominal X nominal (gender and voting preferences)
Nominal and ordinal (gender and opinion for W)

12. X2 can also be used with larger tables
Opinion of Bush
MALE
FEMALE
Favorable 40 5
Indifferent 10 20
Unfavorable 15 55 45
(19.4)
(15.8) 30
(.88)
(.72) 70
(8.6)
(6.9) 65 80
145
X2= Do we reject the null hypothesis?

13. Correlation (Does not mean causation)
We want to know how two variables are related to each other
Does eating doughnuts affect weight?
Does spending more hours studying increase test scores?
Correlation means how much two variables overlap with each other

14. Types of Correlations X (cause) Y (effect) Correlation Values
Increases
Positive
0 to1
Decreases
0 to 1
Negative
-1 to 0
Increase
Does not change
Independent

15. Conceptualizing Correlation
Measuring Development
Strong
Weak
GPD
POP WEIGHT
GDP
EDUCATION
Correlation will be associated with what type of validity?

16. Correlation Coefficient

17. Home Value & Square footage
Log value
Log sqft
value2
sqft2
Val * sqft
5.13
4.02
5.2
4.54
27.04
23.608
4.53
3.53
4.79
3.8
14.44
18.202
4.78
3.86
4.72
4.17
29.15
23.92
141.95
95.96
116.56

18. Correlation Coefficient

19. Rules of Thumb Size of correlation coefficient General Interpretation
Very Strong
Strong
Moderate
Weak
Very Weak or no relationship

20. Multiple Correlation Coefficients

21. Limitation of correlation coefficients
They tell us how strong two variables are related
However, r coefficients are limited because they cannot tell anything about:
Causation between X and Y
Marginal impact of X on Y
What percentage of the variation of Y is explained by X
Forecasting
Because of the above Ordinary Least Square (OLS) is most useful

22. Do you have the BLUES?
B for Best (Minimum error)
L for Linear (The form of the relationship)
U for Un-bias (does the parameter truly reflect the effect?)
E for Estimator

23. Home value and sq. Feet
Does the above line meet the BLUE criteria?