chapter 15 the chi-square statistic: tests for goodness of fit and independence powerpoint lecture...
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![Page 1: Chapter 15 The Chi-Square Statistic: Tests for Goodness of Fit and Independence PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences](https://reader035.vdocuments.mx/reader035/viewer/2022062309/5697c01f1a28abf838cd19d7/html5/thumbnails/1.jpg)
Chapter 15The Chi-Square Statistic: Tests for Goodness of Fit and Independence
PowerPoint Lecture Slides
Essentials of Statistics for the Behavioral Sciences Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
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Chapter 15 Learning Outcomes
•Explain when chi-square test is appropriate
1
•Test hypothesis about shape of distribution using chi-square goodness of fit
2
•Test hypothesis about relationship of variables using chi-square test of independence
3
•Evaluate effect size using phi coefficient or Cramer’s V
4
![Page 3: Chapter 15 The Chi-Square Statistic: Tests for Goodness of Fit and Independence PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences](https://reader035.vdocuments.mx/reader035/viewer/2022062309/5697c01f1a28abf838cd19d7/html5/thumbnails/3.jpg)
Tools You Will Need
• Proportions (math review, Appendix A)
• Frequency distributions (Chapter 2)
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15.1 Parametric and Nonparametric Statistical Tests
• Hypothesis tests used thus far tested hypotheses about population parameters
• Parametric tests share several assumptions– Normal distribution in the population– Homogeneity of variance in the population– Numerical score for each individual
• Nonparametric tests are needed if research situation does not meet all these assumptions
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Chi-Square and Other Nonparametric Tests
• Do not state the hypotheses in terms of a specific population parameter
• Make few assumptions about the population distribution (“distribution-free” tests)
• Participants usually classified into categories– Nominal or ordinal scales are used– Nonparametric test data may be frequencies
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Circumstances Leading to Use of Nonparametric Tests
• Sometimes it is not easy or possible to obtain interval or ratio scale measurements
• Scores that violate parametric test assumptions• High variance in the original scores• Undetermined or infinite scores cannot be
measured—but can be categorized
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15.2 Chi-Square Test for “Goodness of Fit”
• Uses sample data to test hypotheses about the shape or proportions of a population distribution
• Tests the fit of the proportions in the obtained sample with the hypothesized proportions of the population
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Figure 15.1 Grade Distribution for a Sample of n = 40 Students
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Goodness of Fit Null Hypothesis
• Specifies the proportion (or percentage) of the population in each category
• Rationale for null hypotheses:– No preference (equal proportions) among
categories, OR– No difference in specified population from the
proportions in another known population
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Goodness of Fit Alternate Hypothesis
• Could be as terse as “Not H0”• Often equivalent to “…population proportions
are not equal to the values specified in the null hypothesis…”
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Goodness of Fit Test Data
• Individuals are classified (counted) in each category, e.g., grades; exercise frequency; etc.
• Observed Frequency is tabulated for each measurement category (classification)
• Each individual is counted in one and only one category (classification)
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Expected Frequencies in the Goodness of Fit Test
• Goodness of Fit test compares the Observed Frequencies from the data with the Expected Frequencies predicted by null hypothesis
• Construct Expected Frequencies that are in perfect agreement with the null hypothesis
• Expected Frequency is the frequency value that is predicted from H0 and the sample size; it represents an idealized sample distribution
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e
eo
f
ff 22 )(
Chi-Square Statistic
• Notation– χ2 is the lower-case Greek letter Chi– fo is the Observed Frequency
– fe is the Expected Frequency
• Chi-Square Statistic
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Chi-Square Distribution• Null hypothesis should
– Not be rejected when the discrepancy between the Observed and Expected values is small
– Rejected when the discrepancy between the Observed and Expected values is large
• Chi-Square distribution includes values for all possible random samples when H0 is true– All chi-square values ≥ 0.
– When H0 is true, sample χ2 values should be small
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Chi-SquareDegrees of Freedom
• Chi-square distribution is positively skewed• Chi-square is a family of distributions
– Distributions determined by degrees of freedom– Slightly different shape for each value of df
• Degrees of freedom for Goodness of Fit Test– df = C – 1– C is the number of categories
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Figure 15.2 The Chi-Square Distribution Critical Region
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Figure 15.3 Chi-square Distributions with Different df
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Locating the Chi-Square Distribution Critical Region
• Determine significance level (alpha)• Locate critical value of chi-square in a
table of critical values according to– Value for degrees of freedom (df)– Significance level chosen
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Figure 15.4Critical Region for Example 15.1
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In the Literature
• Report should describe whether there were significant differences between category preferences
• Report should include– χ2 df, sample size (n) and test statistic value– Significance level
• E.g., χ2 (3, n = 50) = 8.08, p < .05
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“Goodness of Fit” Test and the One Sample t Test
• Nonparametric versus parametric test• Both tests use data from one sample to test a
hypothesis about a single population• Level of measurement determines test:
– Numerical scores (interval / ratio scale) make it appropriate to compute a mean and use a t test
– Classification in non-numerical categories (ordinal or nominal scale) make it appropriate to compute proportions or percentages to do a chi-square test
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Learning Check
• The expected frequencies in a chi-square test _____________________________________
•are always whole numbers
A
•can contain fractions or decimal values
B
•can contain both positive and negative values
C
•can contain fractions and negative numbers
D
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Learning Check - Answer
• The expected frequencies in a chi-square test _____________________________________
•are always whole numbers
A
•can contain fractions or decimal values
B
•can contain both positive and negative values
C
•can contain fractions and negative numbers
D
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Learning Check
• Decide if each of the following statements is True or False
•In a Chi-Square Test, the Observed Frequencies are always whole numbers
T/F
•A large value for Chi-square will tend to retain the null hypothesis
T/F
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Learning Check - Answers
•Observed frequencies are just frequency counts, so there can be no fractional values
True
•Large values of chi-square indicate observed frequencies differ a lot from null hypothesis predictions
False
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15.3 Chi-Square Test for Independence
• Chi-Square Statistic can test for evidence of a relationship between two variables– Each individual jointly classified on each variable– Counts are presented in the cells of a matrix– Design may be experimental or non-experimental
• Frequency data from a sample is used to test the evidence of a relationship between the two variables in the population using a two-dimensional frequency distribution matrix
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Null Hypothesis for Test of Independence
• Null hypothesis: the two variables are independent (no relationship exists)
• Two versions– Single population: No relationship between two
variables in this population.– Two separate populations: No difference between
distribution of variable in the two populations• Variables are independent if there is no
consistent predictable relationship
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Observed and Expected Frequencies
• Frequencies in the sample are the Observed frequencies for the test
• Expected frequencies are based on the null hypothesis prediction of the same proportions in each category (population)
• Expected frequency of any cell is jointly determined by its column proportion and its row proportion
![Page 29: Chapter 15 The Chi-Square Statistic: Tests for Goodness of Fit and Independence PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences](https://reader035.vdocuments.mx/reader035/viewer/2022062309/5697c01f1a28abf838cd19d7/html5/thumbnails/29.jpg)
Computing Expected Frequencies
• Frequencies computed by same method for each cell in the frequency distribution table
– fc is frequency total for the column
– fr is frequency total for the row
n
fff rce
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Computing Chi-Square Statistic for Test of Independence
• Same equation as the Chi-Square Test of Goodness of Fit
• Chi-Square Statistic• Degrees of freedom (df) = (R-1)(C-1)
– R is the number of rows– C is the number of columns
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Chi-Square Compared to Other Statistical Procedures
• Hypotheses may be stated in terms of relationships between variables (version 1) or differences between groups (version 2)
• Chi-square test for independence and Pearson correlation both evaluate the relationships between two variables
• Depending on the level of measurement, Chi-square, t test or ANOVA could be used to evaluate differences between various groups
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Chi-Square Compared to Other Statistical Procedures (cont.)
• Choice of statistical procedure determined primarily by the level of measurement
• Could choose to test the significance of the relationship– Chi-square– t test– ANOVA
• Could choose to evaluate the strength of the relationship with r2
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15.4 Measuring Effect Sizefor Chi-Square
• A significant Chi-square hypothesis test shows that the difference did not occur by chance– Does not indicate the size of the effect
• For a 2x2 matrix, the phi coefficient (Φ) measures the strength of the relationship
• So Φ2 would provide proportion of variance accounted for just like r2 n
2
φ
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Effect size in a larger matrix
• For a larger matrix, a modification of the phi-coefficient is used: Cramer’s V
• df* is the smaller of (R-1) or (C-1)
*)(
2
dfnV
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Interpreting Cramer’s V
Small Effect
Medium Effect
Large Effect
For df* = 1 0.10 0.30 0.50For df* = 2 0.07 0.21 0.35For df* = 3 0.06 0.17 0.29
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15.5 Chi-Square Test Assumptions and Restrictions
• Independence of observations
– Each observed frequency is generated by adifferent individual
• Size of expected frequencies– Chi-square test should not be performed when the
expected frequency of any cell is less than 5
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Learning Check
• A basic assumption for a chi-square hypothesis test is ______________________
•the population distribution(s) must be normal
A
•the scores must come from an interval or
ratio scale
B
•the observations must be independent
C
•None of the other choices are assumptions
for chi-square
D
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Learning Check - Answer
• A basic assumption for a chi-square hypothesis test is ______________________
•the population distribution(s) must be normal
A
•the scores must come from an interval or
ratio scale
B
•the observations must be independent
C
•None of the other choices are assumptions
for chi-square
D
![Page 39: Chapter 15 The Chi-Square Statistic: Tests for Goodness of Fit and Independence PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences](https://reader035.vdocuments.mx/reader035/viewer/2022062309/5697c01f1a28abf838cd19d7/html5/thumbnails/39.jpg)
Learning Check
• Decide if each of the following statements is True or False
•The value of df for a chi-square test does not depend on the sample size (n)
T/F
•A positive value for the chi-square statistic indicates a positive correlation between the two variables
T/F
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Learning Check - Answers
•The value of df for a chi-square test depends only on the number of rows and columns in the observation matrix
True
•Chi-square cannot be a negative number, so it cannot accurately show the type of correlation between the two variables
False
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Figure 15.5: Example 15.2 SPSS Output—Chi-square Test for Independence
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AnyQuestions
?
Concepts?
Equations?