Chi Square & Correlation

Nonparametric Test of Chi2

Used when too many assumptions are violated in T-Tests:

Sample size too small to reflect populationData 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

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

2525

2525Voted for Bush 50

Voted for Kerry 50

1005050

NOTE: INDEPENDENT VARIABLES ON COLUMS AND DEPENDENT ON ROWS

Calculating X2

What would a contingency table look like if a perfect relationship exists between gender and voting for Bush?

Male Female

500

050Voted for Bush

Voted for Kerry

Calculating the expected value

Nff

f jiij))((^

=

=ijf

^

The expected frequency of the cell in the ith row and jth column

Fi = The total in the ith row marginalFj = The total in the jth column marginalN = The grand total, or sample size for the entire table

Expected Voted for Bush = 50x50 / 100 = 25

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?

∑ −= EEO 22 )(χ

O= Observed frequency E= Expected frequency

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.

Let’s do a X2 (50-25)2/25=25 (0 - 25)2 /25=25(0 - 25)2 /25=25(50-25)2 /25=25

X2=100

Male Female

500

050Voted forBush Voted For Kerry

What would X2 be when there is statistical independence?

Let’s corroborate with SPSS Chi-Square Tests

.000b 1 1.000

.000 1 1.000

.000 1 1.0001.000 .579

.000 1 1.000

100

Pearson Chi-SquareContinuity Correction a

Likelihood RatioFisher's Exact TestLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)Exact Sig.(2-sided)

Exact Sig.(1-sided)

Computed only for a 2x2 tablea.

0 cells (.0%) have expected count less than 5. The minimum expected count is25.00.

b.

Chi-Square Tests

100.000b 1 .00096.040 1 .000

138.629 1 .000.000 .000

99.000 1 .000

100

Pearson Chi-SquareContinuity Correction a

Likelihood RatioFisher's Exact TestLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)Exact Sig.(2-sided)

Exact Sig.(1-sided)

Computed only for a 2x2 tablea.

0 cells (.0%) have expected count less than 5. The minimum expected count is25.00.

b.

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 X2distribution to look for the critical value (CV= 3.84)We conclude that the relationship gender and voting is statistically significant.

Male Female

2030

3020Voted forBush Voted forKerry

X2= 4

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)

X2 can also be used with larger tables

5515Unfavorable

2010Indifferent

540Favorable

FEMALEMALEOpinion of Bush

45(15.8)(19.4)

30(.72)(.88)

70(8.6) (6.9)65 14580

X2=52.3 Do we reject the null hypothesis?

Correlation (Does not mean causation)

We want to know how two variables are related to each otherDoes eating doughnuts affect weight? Does spending more hours studying increase test scores? Correlation means how much two variables overlap with each other

Types of Correlations

-1 to 0NegativeIncreasesDecreases

ValuesCorrelationY (effect)X (cause)

0 IndependentDoes not change

IncreaseDecreases

-1 to 0NegativeDecreasesIncreases

0 to 1 PositiveDecreasesDecreases

0 to1 PositiveIncreasesIncreases

Conceptualizing Correlation

Measuring Development StrongWeak

GPD POP WEIGHT GDP EDUCATION

Correlation will be associated with what type of validity?

Correlation Coefficient

])(][)([ 2222 YYnXXn

YXXYnrxy∑−∑∑−∑

∑∑−∑=

Home Value & Square footage

116.5695.96141.9523.9229.15

19.682417.388922.27844.174.72

18.450814.899622.84843.864.78

18.20214.4422.94413.84.79

15.990912.460920.52093.534.53

23.60820.611627.044.545.2

20.622616.160426.31694.025.13

Val * sqftsqft2value2Log sqftLog value

Correlation Coefficient

])92.23()6*96.95[(])15.29()6*95.141[()92.23)(15.29()56.116*6(

22 −−

−=xyr

Correlations

1 .778. .068

6 6.778 1.068 .

6 6

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)N

VALUE

SQFT

VALUE SQFT

66.209.278. =

Rules of Thumb

Very Weak or no relationship

.0 - .2

Weak.2 - .4

Moderate.4 - .6

Strong .6 - .8

Very Strong.8 - 1.0

General InterpretationSize of correlation coefficient

Multiple Correlation Coefficients

Correlations

1 .784** .775** .708**. .000 .000 .000

46 46 46 46.784** 1 .669** .654**.000 . .000 .000

46 46 46 46.775** .669** 1 .895**.000 .000 . .000

46 46 46 46.708** .654** .895** 1.000 .000 .000 .

46 46 46 46

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)N

VALUE

SQFT

BTH

BDR

VALUE SQFT BTH BDR

Correlation is significant at the 0.01 level (2-tailed).**.

Limitation of correlation coefficients

They tell us how strong two variables are relatedHowever, r coefficients are limited because they cannot tell anything about:

1. Causation between X and Y 2. Marginal impact of X on Y 3. What percentage of the variation of Y is explained

by X 4. Forecasting Because of the above Ordinary Least Square (OLS) is

most useful

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

Home value and sq. Feet

SQFT

4.64.44.24.03.83.63.4

VALU

E

5.3

5.2

5.1

5.0

4.9

4.8

4.7

4.6

4.5

εβα ++= XY

Does the above line meet the BLUE criteria?

Chi Square & CorrelationNonparametric Test of Chi2Calculating X2Calculating X2Calculating the expected valueNonparametric Test of Chi2Nonparametric Test of Chi2Let’s do a X2Let’s corroborate with SPSSTesting for significanceWhen is X2 appropriate to use?X2 can also be used with larger tablesCorrelation (Does not mean causation)Types of CorrelationsConceptualizing CorrelationCorrelation CoefficientHome Value & Square footageCorrelation CoefficientRules of ThumbMultiple Correlation CoefficientsLimitation of correlation coefficientsDo you have the BLUES?Home value and sq. Feet