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Chapter 14: Chi-Square

Introduction to the Chi‐Square Statistic

• In Chapters 9, the research situations consisted of categorical IV(s) and continuous DV

• In this chapter, we discuss research situations where all of the variables are categorical

• In Chapter 13, we discuss research situations where both the IV and DV were continuous

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/20/2018

• Chi‐square addresses the question,

– Are the frequencies observed in a sample significantly different from frequencies we expect?

– Expected frequencies may come from • Previous theory or research

• Assumption of equal distributions among the categories

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

• Characteristics of the 2 distribution – Shape of distribution differs depending on df

– Because it’s based on (fo – fe) 2 , 2 is always a

positive number

– 2 distribution is positively skewed

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

12/20/2018

• Assumptions of the 2 distribution – Independence of observations

• No observation may be in more than one category

– Minimum expected frequencies • none of the expected frequencies should be zero (0)

• most if not all of the expected frequencies should be > 5

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

• Calculate a statistic: Chi‐square (2) ‐ Calculate expected frequencies (fe)

fe = (hypothesized proportion) (N)

‐ Calculate the 2 statistic

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• An example: Researchers categorized participants into one of four personality types

– Alphas (extraverted and norm‐favoring)

– Betas (introverted and norm‐favoring)

– Gammas (extraverted and norm‐questioning)

– Deltas (introverted and norm‐questioning)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

• Are the four personality types equally observed in a sample of 588 students?

Introduction to the Chi‐Square Statistic

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/20/2018

Inferential Statistics: Chi‐Square Goodness of Fit Test

• When we have one categorical variable, we test the differences between the observed and expected frequencies using the chi‐square goodness of fit test

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

12/20/2018

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Introduction to the Chi‐Square Statistic

12/20/2018

• State the null and alternative hypotheses (H0 and H1)

– H0: distribution of observed frequencies fits the distribution of expected frequencies

– H1: distribution of observed frequencies does not fit the distribution of expected frequencies

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

• Make a decision about the null hypothesis – Calculate the degrees of freedom (df)

– Set alpha (α), identify the critical value, and state a decision rule

– Calculate a statistic: Chi‐square (2) – Make a decision whether to reject the null hypothesis

– Determine the level of significance; and

– Calculate a measure of effect size (Cramér's )

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

12/20/2018

Inferential Statistics: Chi‐Square Goodness of Fit Test

– Calculate the degrees of freedom (df)

df = # groups – 1

= 4 – 1

= 3

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

– Set alpha (α), identify the critical value, and state a decision rule

• For  = .05 and df = 3, critical value = 7.81 • If 2 > 7.81, reject H0; otherwise, do not reject H0

Inferential Statistics: Chi‐Square Goodness of Fit Test

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/20/2018

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

• Calculate a statistic: Chi‐square (2) ‐ Calculate expected frequencies (fe)

fe = (hypothesized proportion) (N)

‐ Calculate the 2 statistic

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

12/20/2018

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

Make a decision whether to reject the null hypothesis

‐ 2 = 166.74 > 7.81  reject H0 (p < .05)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

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– Determine the level of significance

– 2 = 166.74 > 11.34  p < .01

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

12/20/2018

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

• Draw a conclusion from the analysis – The distribution of the four personality types in the sample of 588 students (Alpha (f = 251 (42.7%)), Beta (f = 65 (11.0%)), Gamma (f = 194 (33.0%)), Delta (f = 78 (13.3%)) was significantly different from the CPI developers’ expected distribution of f = 147 (25%) for each of the four types, 2(3, N = 588) = 166.74, p < .01,  = .30.

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

12/20/2018

• Relate the result of the analysis to the research hypothesis

– The result of this analysis supports the research hypothesis that the distribution of personality types among college students does not match or fit the distribution in the population proposed by the developers of the CPI.

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

• What about situations with unequal expected frequencies?

– Based on previous research, what if we expected the following?

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

12/20/2018

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Goodness of Fit Test

12/20/2018

• The chi‐square test for independence allows us to test two nominal variables

• Research example – Children were presented a picture of either an angry or happy face, then given the opportunity to select a toy of a different color (red, green, or gray). Each child’s color preference was recorded

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Test of Independence

• Contingency table

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Test of Independence

12/20/2018

• Bar chart

Inferential Statistics: Chi‐Square Test of Independence

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Chi‐Square Test of Independence

• State the null and alternative hypotheses (H0 and H1)

– H0: the two variables are independent (i.e., there is no relationship between the two variables)

– H1: the two variables are not independent (i.e., there is a relationship between the two variables)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/20/2018

• Make a decision about the null hypothesis – Calculate the degrees of freedom (df)

– Set alpha (α), identify the critical value, and state a decision rule

– Calculate a statistic: Chi‐square (2) – Make a decision whether to reject the null hypothesis

– Determine the level of significance; and

– Calculate a measure of effect size (Cramér's )

Inferential Statistics: Chi‐Square Test of Independence

– Calculate the degrees of freedom (df)

df = (# rows – 1)(# columns – 1)

= (3 – 1)(2 – 1) = (2)(1)

= 2

Inferential Statistics: Chi‐Square Test of Independence

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– Set alpha (α), identify the critical value, and state a decision rule

‐ For  = .05 and df = 2, critical value = 5.99 ‐ If 2 > 5.99, reject H0; otherwise, do not reject H0

Inferential Statistics: Chi‐Square Test of Independence

• Calculate a statistic: Chi‐square (2) − Calculate expected frequencies

12/20/2018

Inferential Statistics: Chi‐Square Test of Independence

• Calculate a statistic: Chi‐square (2) − Calculate the chi‐square statistic

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Inferential Statistics: Chi‐Square Test of Independence

– Make a decision whether to reject the null hypothesis

• 2 = 8.24 > 5.99  reject H0 (p < .05)

Inferential Statistics: Chi‐Square Test of Independence

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– Determine the level of significance • 2 = 8.24 < 9.21  p < .05 (but not < .01)

Inferential Statistics: Chi‐Square Test of Independence

12/20/2018

• Draw a conclusion from the analysis – The frequencies representing the color preferences (red, green, or gray) of 40 infants shown either a happy or angry face were analyzed using the chi‐square test of independence. This analysis found a significant relationship between Toy color and Face such that the distribution of the three toy colors is different for infants shown the happy face versus the angry face, 2(2, N = 40) = 8.24, p < .05,  = .46.

Inferential Statistics: Chi‐Square Test of Independence

• Relate the result of the analysis to the research hypothesis

– “Our findings indicate that infants’ preference for red changes with the context in which it is presented. Specifically, in a hospitable context, red is preferred, whereas in a hostile context, red is not preferred” (Maier et al 2009, p. 737).

Inferential Statistics: Chi‐Square Test of Independence

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Parametric and Nonparametric Statistical Tests

Parametric Tests

• Assumptions: – Tests used to estimate population parameters

– Populations based on normal distributions

– Samples also normally distributed

Nonparametric Tests

• Features:

– Not estimating population parameters

– No assumption of normality

– “distribution free”

• Why use nonparametric tests? – Data are either nominal or ordinal in measurement type

– Data are interval or ratio level, but are not normally distributed

Parametric and Nonparametric Statistical Tests

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Looking Ahead

• This chapter examined research situations in which all of the variables in the research studies were categorical in nature.

• Next: Chapter 13 and Ordinary Least Squares regression for continuous variables.