Week 13a

Week 13aWeek 13a

Making Inferences, Part IIIMaking Inferences, Part III

t t andand chi-square testschi-square tests

POLS/GEOG 418 Spring 2005POLS/GEOG 418 Spring 2005 22

Lecture OutlineLecture Outline

Review of Review of tt tests and tests and pp values valuesCalculating precise probabilitiesCalculating precise probabilitiesHypothesis tests for nominal and ordinal Hypothesis tests for nominal and ordinal

variablesvariables


Review: confidence Review: confidence intervalsintervals

We can construct an interval containing 95 We can construct an interval containing 95 percent of the observationspercent of the observationsCalculate LCB and UCBCalculate LCB and UCBUsing mean, s.e. and Using mean, s.e. and ss

Useful for hypothesis testingUseful for hypothesis testing ““confidence” in our findingsconfidence” in our findings


Review: confidence Review: confidence intervalsintervals

Sample size matters!Sample size matters!Large samples: we assume normal Large samples: we assume normal

distributiondistributionnn >= 1,000 >= 1,000

Small samples: use the Small samples: use the tt distribution distributionnn < 1,000 < 1,000Look up values in McClendonLook up values in McClendon


Review: Review: tt tests tests

Two hypotheses for means comparisonsTwo hypotheses for means comparisonsDoes the sample mean differ from a Does the sample mean differ from a

hypothesized value?hypothesized value?Independent-sample Independent-sample tt test test



Two hypotheses for means comparisonsTwo hypotheses for means comparisonsDo the means for two groups in the sample Do the means for two groups in the sample

differ from each other?differ from each other?Two-sample Two-sample tt test test



Independent-sampleIndependent-sample If hypothesized value is outside the C.I., we If hypothesized value is outside the C.I., we

reject the alternate hypothesisreject the alternate hypothesis

Two-sampleTwo-sample If the C.I. for the hypothesized difference If the C.I. for the hypothesized difference

contains zero, we reject the alternativecontains zero, we reject the alternative


Review: Review: pp values values

Alternative to Alternative to tt tests testsMore precise: “precise probability”More precise: “precise probability”Assigns any observation a probability Assigns any observation a probability pp

that we would observe it by chancethat we would observe it by chance If If pp is low (less than .05) we accept the is low (less than .05) we accept the

alternative hypothesisalternative hypothesis


Review: SPSS outputReview: SPSS output

Independent-sample Independent-sample tt test test

One-Sample Test

72.435 713 .000 12.8935 12.5441 13.2430SSTRANt df Sig. (2-tailed)

MeanDifference Lower Upper

95% ConfidenceInterval of the

Difference

Test Value = 0

ConfidenceIntervalp value


Review: SPSS outputReview: SPSS output

Two-sample Two-sample tt test test

Independent Samples Test

30.743 .000 15.648 712 .000 5.5170 .35257 4.82481 6.20923

19.265 491.843 .000 5.5170 .28637 4.95436 6.07968

Equal variancesassumed

Equal variancesnot assumed

SSTRANF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper


Difference

t-test for Equality of Means

ConfidenceIntervalp value


ReviewReview

Questions?Questions?


Obtaining Obtaining pp values values

In SPSSIn SPSSManuallyManually



A simple test statisticA simple test statisticThe difference of the hypothesized mean The difference of the hypothesized mean

and the null mean, divided by its standard and the null mean, divided by its standard errorerror

difference of ..statisticTest

es

HH OA



Standard error of differenceStandard error of difference

22

21 ...... meanmeandiff eseses


Obtaining Obtaining p p valuesvalues

The test statistic . . .The test statistic . . . . . . is normally distributed for large samples. . . is normally distributed for large samples . . . is normally distributed when the . . . is normally distributed when the

population variance is knownpopulation variance is known . . . follows the . . . follows the tt distribution when sample size distribution when sample size

is small, or when we don’t know the is small, or when we don’t know the population variancepopulation variance



Example:Example: Do European governments Do European governments spend more on social welfare than non-spend more on social welfare than non-European governments?European governments?

HHaa: “In comparing governments, those in : “In comparing governments, those in

Europe will spend more on social welfare Europe will spend more on social welfare than those outside of Europe.”than those outside of Europe.”



Example:Example: social welfare spending in social welfare spending in Europe and outsideEurope and outside

Group Statistics

532 14.2998 4.45284 .19305

182 8.7828 2.85345 .21151

European dummyEuropean state

Non-European state

SSTRANN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

5.5170 .35257 4.82481 6.20923

5.5170 .28637 4.95436 6.07968

Equal variancesassumed

Equal variancesnot assumed

SSTRAN

MeanDifference

Std. ErrorDifference Lower Upper


Difference

t-test for Equality of Means



Example:Example: social welfare spending in social welfare spending in Europe and outsideEurope and outsideAssume a large sample Assume a large sample

23.19286.0

8.83.14..

es

HHZ OA



Example:Example: social welfare spending in social welfare spending in Europe and outsideEurope and outsideAssume a small sampleAssume a small sample

23.19286.0

8.83.14..

es

HHt OA



Example:Example: social welfare spending in social welfare spending in Europe and outsideEurope and outside Is 19.23 a significant test statistic?Is 19.23 a significant test statistic?



Three ways to get Three ways to get p p value for a given value for a given tt or or ZZEyeball testEyeball testStudent’sStudent’s t t table tableSPSS or ExcelSPSS or Excel



Eyeball testEyeball test Is your test statistic (Is your test statistic (ZZ or or tt) greater than two?) greater than two? If so, you can reject the null and accept the If so, you can reject the null and accept the

alternative hypothesisalternative hypothesis19.23 is far greater than two, so we accept 19.23 is far greater than two, so we accept

the hypothesis that Europe spends more on the hypothesis that Europe spends more on social welfaresocial welfare



Student’s Student’s tt table tableLook up critical Look up critical tt value in a table value in a table

NOTE:NOTE: degrees of freedom = degrees of freedom = nn – 1 – 1

If your statistic exceeds the critical value, If your statistic exceeds the critical value, accept the alternative hypothesisaccept the alternative hypothesis



Student’s Student’s tt table table



From SPSSFrom SPSS


Limitations of Limitations of ZZ, , tt and and pp

Great for means comparisonsGreat for means comparisonsCannot use with nominal or ordinal Cannot use with nominal or ordinal

variablesvariablesSince zero has no meaningSince zero has no meaning


Chi-square testChi-square test

Used forUsed forNominal variablesNominal variablesOrdinal variablesOrdinal variables

In conjunction with cross tabulationIn conjunction with cross tabulation



Recall example from last TuesdayRecall example from last TuesdayHHaa: “In comparing voters, those with more : “In comparing voters, those with more

education will favor tougher environmental education will favor tougher environmental regulations than those with less education.”regulations than those with less education.”



Recall:Recall:Environmental regulations:3 cats * Education: 3 categories Crosstabulation

44 169 445 658

46.8% 48.4% 49.1% 48.8%

34 116 325 475

36.2% 33.2% 35.9% 35.2%

16 64 136 216

17.0% 18.3% 15.0% 16.0%

94 349 906 1349

100.0% 100.0% 100.0% 100.0%

Count

% within Education:3 categories

Count


Count


Count


Tougher regs

Depends

Regs a burden

Environmentalregulations:3cats

Total

1. Lessthan HS 2. HS

3. Morethan HS

Education: 3 categories

Total

49.1% - 46.8% = 2.3%



Intuition: Intuition: what percentage would we what percentage would we expect expect to see in to see in

each cell if there is no relationship?each cell if there is no relationship?

Chi-square test measures the differences Chi-square test measures the differences between between observed observed and and expectedexpected frequencies in each cellfrequencies in each cell


Expected frequenciesExpected frequencies

MenMen WomenWomen TotalTotal

YesYes ?? ?? 100100

NoNo ?? ?? 100100

TotalTotal 100100 100100 200200




YesYes 5050 5050 100100

NoNo 5050 5050 100100

TotalTotal 100100 100100 200200




YesYes ?? ?? 4040

NoNo ?? ?? 160160

TotalTotal 100100 100100 200200




YesYes 2020 2020 4040

NoNo 8080 8080 160160

TotalTotal 100100 100100 200200



If there is no relationship, we expect each If there is no relationship, we expect each cell cell within a categorywithin a category will follow the same will follow the same proportion as the overall sampleproportion as the overall sample




YesYes ?? ?? 100100

NoNo ?? ?? 100100

TotalTotal 100100 100100 200200

fe = (100 / 200) * 100 = 50

OverallProportion:

100 yeses outOf 200 total




YesYes ?? 5050 100100

NoNo ?? ?? 100100

TotalTotal 100100 100100 200200



For any crosstab, we know For any crosstab, we know the totals for each value of the dependent the totals for each value of the dependent

variable (rows)variable (rows) the totals for each group of the independent the totals for each group of the independent

variable (columns)variable (columns)We can calculate expected frequencies We can calculate expected frequencies

for any cell in any tablefor any cell in any table



The sum of squared differences between The sum of squared differences between observed and expected frequencies, observed and expected frequencies, divided by the expected frequency, follows divided by the expected frequency, follows the the chi-square distributionchi-square distribution

e

eo

f

ff 22



In six stepsIn six steps1.1. Find the expected frequency for each cellFind the expected frequency for each cell

2.2. Subtract the expected from the observed Subtract the expected from the observed frequency in each cellfrequency in each cell

3.3. For each cell, square the figure you obtained For each cell, square the figure you obtained in step #2in step #2

4.4. Divide this figure by Divide this figure by ffee

5.5. Add up all the totals from step 4Add up all the totals from step 4

6.6. Look up critical values in a chi-square tableLook up critical values in a chi-square table



SPSS example: Do liberals favor gun SPSS example: Do liberals favor gun control more than conservatives?control more than conservatives?

HHaa: “In comparing voters, liberals will : “In comparing voters, liberals will

express stronger support for gun control express stronger support for gun control than will conservatives”than will conservatives”



SPSS output, without the test statisticSPSS output, without the test statistic

Favor or Oppose Gun Permits * Politicial Orientation Crosstabulation

228 289 269 786

90.1% 81.4% 79.1% 82.9%

25 66 71 162

9.9% 18.6% 20.9% 17.1%

253 355 340 948

100.0% 100.0% 100.0% 100.0%

Count

% within PoliticialOrientation

Count


Count


Favor

Oppose

Favor or OpposeGun Permits

Total

Liberal Independent Conservative

Politicial Orientation

Total



In SPSS:In SPSS:



In SPSSIn SPSS



SPSS outputSPSS outputFavor or Oppose Gun Permits * Politicial Orientation Crosstabulation

228 289 269 786

209.8 294.3 281.9 786.0

90.1% 81.4% 79.1% 82.9%

25 66 71 162

43.2 60.7 58.1 162.0

9.9% 18.6% 20.9% 17.1%

253 355 340 948

253.0 355.0 340.0 948.0

100.0% 100.0% 100.0% 100.0%

Count

Expected Count


Count

Expected Count


Count

Expected Count


Favor

Oppose


Total



Total



In SPSSIn SPSS



SPSS outputSPSS outputFavor or Oppose Gun Permits * Politicial Orientation Crosstabulation

228 289 269 786

209.8 294.3 281.9 786.0

90.1% 81.4% 79.1% 82.9%

25 66 71 162

43.2 60.7 58.1 162.0

9.9% 18.6% 20.9% 17.1%

253 355 340 948

253.0 355.0 340.0 948.0

100.0% 100.0% 100.0% 100.0%

Count

Expected Count


Count

Expected Count


Count

Expected Count


Favor

Oppose


Total



Total

Chi-Square Tests

13.295a 2 .001

14.447 2 .001

11.681 1 .001

948

Pearson Chi-Square

Likelihood Ratio

Linear-by-LinearAssociation

N of Valid Cases

Value dfAsymp. Sig.

(2-sided)

0 cells (.0%) have expected count less than 5. Theminimum expected count is 43.23.

a.

Week 13a

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