Biology - Anatomy Assignment 6

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Nonparametric Tests / Chi Square Tests

Overview

  • What are nonparametric tests?
  • Parametric vs. Nonparametric tests
  • Restrictions of nonparametric tests
  • Chi-square test for independence
  • Spearman Rho correlation
  • Other Nonparametric tests

What Are Nonparametric Tests?

  • Nonparametric tests require few, if any assumptions about the shapes of the underlying population distributions
  • For this reason, they are often used in place of parametric tests if or when one feels that the assumptions of the parametric test have been too grossly violated (e.g., if the distributions are too severely skewed).

Parametric or Nonparametric tests?

  • Non-parametric tests typically used when:
  • The dependent variable is either nominal or ordinal

  • A specific research question can only be answered using a non-parametric test
  • Parametric tests are not available for answering some kinds of research questions.

Restrictions

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  • Nonparametric tests do have at least two major disadvantages in comparison to parametric tests:
  • First, nonparametric tests are less powerful. Why? Because parametric tests use more of the information available in a set of numbers.
  • Parametric tests make use of information consistent with interval or ratio scale (or continuous) measurement, whereas nonparametric tests typically make use of nominal or ordinal (or categorical) information only.

Restrictions (cont’d)

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  • Second, parametric tests are much more flexible, and allow you to test a greater range of hypotheses. For example, ANOVA designs allow you to test for interactions between variables in a way that is not possible with nonparametric alternatives.
  • There are nonparametric techniques to test for certain kinds of interactions under certain circumstances, but these are much more limited than the corresponding parametric techniques.

Chi-Square Test

  • Used to test hypotheses concerning the frequency of occurrences for qualitative variables in a dataset.
  • Qualitative variables on the nominal scale:
  • Gender
  • Political affiliation
  • Course delivery method
  • Three main types:
  • Goodness-of-fit (or one-sample) test
  • Test of independence (or association)
  • Independent-samples test

Brief historical pause….

Karl Pearson (b. London1857; d. London 1936)

Developed the chi-square test (published July 1900)

Invented the term “standard deviation”

Invented the product-moment correlation

Invented the Pearson chi-square tests

Chi-Square Goodness-of-Fit Test

  • Evaluates whether the proportions of individuals who fall into categories are equal to hypothesized values
  • The variable can have two or more categories
  • The categories can have quantitative (one category reflects a higher value than another; e.g., Likert scale responses of Agree and Disagree) or qualitative grouping (e.g., gender)
  • Note: The chi-square test does not recognize any quantitative distinction among categories; it simply assesses whether the frequencies of occurrence associated with the categories are significantly different from the hypothesized frequencies of occurrence in the population

Chi-Square Goodness-of-Fit Test(cont’d)

  • When conducting a chi-square test in SPSS, you must first specify the values for the hypothesized frequencies
  • The hypothesized frequencies can be
  • Equal
  • Not equal (determined based on previous research)
  • E.g., suppose you wanted to see if your frequencies for ethnicity categories matched the same proportions as determined by the U.S. Census…

Chi-Square Goodness-of-Fit Test(cont’d)

  • The hypothesized frequencies can be
  • Equal frequencies (add up the totals and divide by number of groups)

Actual Frequencies Hypothesized Frequencies

Group 1

Group 2

Group 3

Total # =

24 28

18 28

42 28

84 84/3 = 28 per group

Goodness of Fit Example

  • You may have heard, “Stay with your first answer on a multiple-choice test.” Is changing answers more likely to be helpful or harmful? To examine this, Best (1979) studied the responses of 261 students in an introductory psychology course. He recorded the number of right-to-wrong and wrong-to-right answer changes for each student. More wrong-to-right changes than right-to-wrong changes were made by 195 of the students, who were thus “helped” by changing answers; 27 students made more right-to-wrong changes than wrong-to-right changes and thus hurt their performance.
  • Are the frequencies of right-to-wrong and wrong-to-right “changers” equal.

Best, J. B. (1979). Item difficulty and answer changing. Teaching of Psychology, 6, 228- 230.

Data = Chi Square Class Example.sav

Goodness-of-Fit Example

  • Step 1: State the null and alternate hypotheses
  • H0 : the frequencies of right-to-wrong and wrong-to-right changers are equal.
  • Ha : The frequencies of right-to-wrong and wrong-to-right changers are NOT equal.

Goodness-of-Fit Example (cont’d)

  • Step 2: Create a contingency table

total f = 222

  • Step 3: Input the data into SPSS
  • Create a variable labeled Answer Change
  • Assign values: 1 = wrong to right, 2 = right to wrong
  • Enter the data (in one column): 195, 1’s and 27, 2’s
Observed Frequencies Hypothesized Proportions Expected Frequencies
More Wrong to Right 195 .50 222*.50 = 111
More Right to Wrong 27 .50 222*.50 = 111

Goodness-of-Fit Example (cont’d)

  • Step 4: Run the Analysis
  • Analyze  Nonparametric  Legacy Dialogs 
    Chi-square
  • Move the variable (DeliveryMethod) to the “Test variable list” box
  • Under “Expected Values,” select “All categories equal”
  • Click “Ok”

Goodness-of-Fit Example (cont’d)

Goodness-of-Fit Example (cont’d)

  • Step 4: Make a decision regarding the null
  • X2 = 127.135
  • df = 1 (Number of categories – 1)
  • p < .001
  • What is our decision regarding the null?

Goodness-of-Fit Example (cont’d)

  • Using the level of significance = .05, do we reject or fail to reject the null?
  • If p < .05, we reject the null
  • if p > .05, we fail to reject the null
  • According to SPSS, p < .001
  • .001 < .05, therefore, we reject the null!

Goodness-of-Fit Example (cont’d)

  • Step 5: Write up your results.
  • The null hypothesis stated that the number of students who make more wrong to right changes on a test are equal to the number of students who make right to wrong changes on a test. A chi-square goodness-of-fit test revealed that significantly more students predominantly made wrong to right changes (n=195) than right to wrong changes (n=27, Χ2 = 127.135, df = 1, p = .001). Consequently, the null hypothesis was rejected.

Some Non-Parametric Correlation Coefficients

  • Spearman’s rho
  • At least one variable is ordinal (the other is ordinal or continuous)
  • Phi
  • Two dichotomous categorical variables
  • Cramer’s C (or V)
  • Two categorical variables with any number of categories

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  • Example: suppose we want to know how strongly teacher’s impressions of their students intelligence are related to students actual intelligence as measured by standard IQ test.

  • Dataset = Pearson vs Spearman Rho class example.sav

Spearman’s rs

1 -

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  • We ask high school freshman to rate self-efficacy for sports performance on scale ranging from 1 (not much) to 6 (a lot); There are 15 students –

  • Self-efficacy not interval or ratio level because no reason to believe that difference in pain between, say, 1 & 2 is same as that between 5 & 6 - Scale is ordinal;

  • Sports performance score is measured at interval/ratio level.

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Spearman Vs Pearson

Nonparametric Tests for Nominal Data

  • The sign test
  • The binomial test
  • Fischer’s Exact Test
  • The chi-square test
  • McNemar’s test
  • Cochran’s test

Nonparametric Tests for Ordinal Data

  • The Mann-Whitney U test
  • Analogous to independent samples t test
  • Wilcoxon signed-ranks test
  • Analogous to paired samples t test
  • The Kruskal-Walis one-way analysis of ranks
  • Analogous to a one-way ANOVA
  • The Friedman’s ANOVA
  • Analogous to RM ANOVA
  • Spearman’s rho
  • Analogous to Pearson correlation