Inferential StatisticsInferential Statistics

Hypothesis testing (relationship Hypothesis testing (relationship between 2 or more variables)between 2 or more variables)

We want to make inferences from a We want to make inferences from a sample to a population.sample to a population.

A random sample allows us to infer from A random sample allows us to infer from a sample to a population.a sample to a population.

Inferential StatisticsInferential Statistics

Significance Tests Significance Tests Z scores (one sample case)Z scores (one sample case) Difference of means tests Difference of means tests

Two sample case (t-test)Two sample case (t-test)

Three or more sample case (ANOVA)Three or more sample case (ANOVA)

Chi-SquareChi-Square Bi-Variate Correlation (One IV & One DV)Bi-Variate Correlation (One IV & One DV) Bi-Variate Regression (One IV & One DV)Bi-Variate Regression (One IV & One DV) Multi-Variate Regression (Two or more IVs & One DV)Multi-Variate Regression (Two or more IVs & One DV)

Level of Measurement & Significance TestsLevel of Measurement & Significance Tests

Chi-Square Chi-Square IV & DV are nominal and/or ordinal IV & DV are nominal and/or ordinal

t-test t-test IV is nominal (group like men & women)IV is nominal (group like men & women)DV is Interval/Ratio (or a scale)DV is Interval/Ratio (or a scale)

ANOVAANOVAIV is nominal (group with 3 or more categories)IV is nominal (group with 3 or more categories)DV is I/R (or a scale)DV is I/R (or a scale)

Regression Regression IV(s) & DV are I/R (or scales) IV(s) & DV are I/R (or scales) IV(s) can be dummy variables IV(s) can be dummy variables

Which Test Would you Use?Which Test Would you Use?

Hr: There is a relationship between:Hr: There is a relationship between: gender & income (measured in dollars)gender & income (measured in dollars) race (measured as Black, Latino/a, race (measured as Black, Latino/a,

Caucasian) and incomeCaucasian) and income religious preference (catholic, religious preference (catholic, protestant) protestant)

and attitudes toward abortion (favor, and attitudes toward abortion (favor, oppose)oppose)

education (measured in years) and incomeeducation (measured in years) and income degree completed (HS or Less & College) degree completed (HS or Less & College)

and incomeand income

Chi-SquareChi-Square

Chi-Square: a test of significance used Chi-Square: a test of significance used with cross tabulations of nominal/ordinal with cross tabulations of nominal/ordinal level data.level data.

Example:Example:

Research question: Does political orientation Research question: Does political orientation influence parenting style?influence parenting style?

Political orientation: Conservative & LiberalPolitical orientation: Conservative & Liberal

Parenting style: Permissive & Not Permissive Parenting style: Permissive & Not Permissive

Why not simply compare the mean difference Why not simply compare the mean difference

between liberals and conservatives on parentingbetween liberals and conservatives on parenting

style?style?

We are really saying:We are really saying:

Hr: The frequency (proportion) of liberals Hr: The frequency (proportion) of liberals who are permissive who are permissive is not the sameis not the same as as the frequency of conservatives who are the frequency of conservatives who are permissive.permissive.

The null (a hypothesis of no difference) The null (a hypothesis of no difference) says:says:

Ho: The frequency (proportion) of liberals Ho: The frequency (proportion) of liberals who are permissive who are permissive is the sameis the same as the as the frequency of conservatives who are frequency of conservatives who are permissive.permissive.

Chi-Square compares the observed Chi-Square compares the observed frequencies (from the data in your frequencies (from the data in your sample) to expected frequencies.sample) to expected frequencies.

Expected frequenciesExpected frequencies: These are the : These are the frequencies we would expect if the null frequencies we would expect if the null were true (if there is no difference were true (if there is no difference between political view and parenting between political view and parenting style)style)

Example:Example:

We do a cross tab of political orientation by We do a cross tab of political orientation by parenting style and our observed frequencies parenting style and our observed frequencies are:are:

Political OrientationPolitical OrientationLiberalsLiberals ConservativesConservatives

Child-rearingChild-rearingPermissivePermissive 5 5 1010Not permissiveNot permissive 1515 1010

______ ______2020 2020

Are these differences significant?Are these differences significant?

Chi-Square test of significance:Chi-Square test of significance:

Chi-Square = Chi-Square = ∑∑((fofo- - fefe)2 / )2 / fefe

StepsSteps

Step 1.Step 1. We have the observed frequenciesWe have the observed frequencies

Political OrientationPolitical Orientation

LiberalsLiberals ConservativesConservatives

Child-rearingChild-rearing

PermissivePermissive 5 5 1010

Not permissiveNot permissive 1515 1010

______ ______

2020 2020

StepsSteps

Step 2. Step 2. Need to calculate the expected frequenciesNeed to calculate the expected frequencies..

Formula:Formula:

fe = (row marginal total) (column marginal total)fe = (row marginal total) (column marginal total)

______________________________________________________________________

NN

Expected FrequenciesExpected Frequencies

See boardSee board

Step 3. Calculate Step 3. Calculate Chi-SquareChi-Square

See boardSee board

Calculated Chi-Square for Political Views by Calculated Chi-Square for Political Views by Parenting StyleParenting Style

Chi Square = 2.66Chi Square = 2.66

Df = (r-1)(c-1)Df = (r-1)(c-1)

Df = (2-1) (2-1) = 1Df = (2-1) (2-1) = 1

Must have a Chi Square of 3.84 at p.=.05Must have a Chi Square of 3.84 at p.=.05to reject the null hypothesis. to reject the null hypothesis.

Decision?Decision?

Review Alpha LevelsReview Alpha Levels

Alpha level the probability of making a Type I Alpha level the probability of making a Type I errorerror

Type I error (reject the null when it is true)Type I error (reject the null when it is true)

Set alpha level small (.05 or smaller) to minimize Set alpha level small (.05 or smaller) to minimize risk.risk.

The larger the sample the smaller the alpha level The larger the sample the smaller the alpha level should be.should be.

Chi square is sensitive to N (large Chi square is sensitive to N (large N’s can yield significant results)N’s can yield significant results)

So, we use a measure of So, we use a measure of association with Chi-squareassociation with Chi-square

Measures of association tell us about Measures of association tell us about thethe

strength of the relationshipstrength of the relationship

Measures of AssociationMeasures of Association

The type of measure used is The type of measure used is determined by the level of measurement determined by the level of measurement and the number of categories.and the number of categories.

See handoutSee handout

Interpret GSS OutputInterpret GSS Output

CrosstabCrosstab

FEELINGS ABOUT PORNOGRAPHY LAWS * HOW FUNDAMENTALIST IS R CURRENTLY Crosstabulation

249 227 150 626

193.2 242.1 190.7 626.0

46.3% 33.7% 28.2% 35.9%

275 431 357 1063

328.1 411.1 323.8 1063.0

51.1% 63.9% 67.2% 61.0%

14 16 24 54

16.7 20.9 16.5 54.0

2.6% 2.4% 4.5% 3.1%

538 674 531 1743

538.0 674.0 531.0 1743.0

100.0% 100.0% 100.0% 100.0%

Count

Expected Count

% within HOWFUNDAMENTALISTIS R CURRENTLY

Count

Expected Count

% within HOWFUNDAMENTALISTIS R CURRENTLY

Count

Expected Count

% within HOWFUNDAMENTALISTIS R CURRENTLY

Count

Expected Count

% within HOWFUNDAMENTALISTIS R CURRENTLY

ILLEGAL TO ALL

ILLEGAL UNDER 18

LEGAL

FEELINGS ABOUTPORNOGRAPHYLAWS

Total

FUNDAMENTALIST MODERATE LIBERAL

HOW FUNDAMENTALIST IS RCURRENTLY

Total

Chi-SquareChi-Square

Chi-Square Tests

43.721a 4 .000

43.149 4 .000

37.689 1 .000

1743

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 16.45.

a.

Measure of AssociationMeasure of Association

Which should we use?Which should we use?

Cramer’s V = .112Cramer’s V = .112