Some nonparametric statistic for categorical & ordinal data

SOME NONPARAMETRIC STATISTIC
FOR CATEGORICAL & ORDINAL DATA
Desmond Ayim-Aboagye, Ph.D.

Preliminary Information
• Chi-square test and others to be studied here
are much less complex than the inferential
tests treated already by us.
• Hypothesis testing but with different sorts of
data
• Nonparametric tests – questions that do not
lend themselves to analysis by the usual
inferential statistics

A Campus Physician
• A campus physician is interested in
determining whether male or female students
are equally aware that regular workshops on
wellness, fitness, and diet are offered by the
health center. She randomly samples 200
students (half male, half female) and asks
whether they are aware of these programs,
subsequently comparing the frequencies of
their responses. The data follows:

Aware Workshops are Offered?
Gender Yes No
Male 30 70
Female 80 20

Frequencies
• Responses are readily categorized as raw
frequencies:
• “Yes, I know about the workshops“ or
• “No, I did not know about the workshops"

Observation
• A nonparametric test performed on such data
enables us to take any speculation a step
further, to consider whether the observed
pattern of responses is statistically valid (i.e.,
women are more aware of available campus
health initiatives than men).

Nonparametric analyses
• 1. Ranks things
• 2. Study association between variables, i.e.,
correlation.

Parametric Tests
• A parametric statistic is an inferential test that, prior to
its use, assumes that certain specific characteristics are
true of a population – particularly the shape of its
distribution – from which data are drawn for analysis.
• Parametric tests concern Population parameters such
as 𝜇, 𝜎 and making inferences about those parameters,
requiring that certain assumptions be met before they
can properly be used for data analysis.
• Bell-shape or normal distributions
• When a set of data are not normal in shape ( i.e., a
distribution is said to be skewed).

Nonparametric Tests
• A nonparametric statistic is an inferential test, one that
makes few or sometimes no assumptions regarding any
numerical data or the shape of the population from
which the observations were drawn.
• Known as “distribution-free tests of significance“
• Normality need not be met.
• Too few observation exist, a distribution is skewed.
• Parametric tests assumptions are violated

Advantages of Using Nonparametric
Tests
• 1. Nonparametric statistical tests are usually
“distribution free“
• 2. Nonparametric statistical tests can be used to
analyze data that are not precisely numerical.[
parametric use only interval and ratio scales
while nonparametric nominally scaled or rank,
that is, ordinally scaled ]
• 3. Nonparametric statistical tests are ideal for
analyzing data from small samples.
• Nonparametric statistical tests are generally easy
to calculate.

Disadvantages of Nonparametric Tests
• 1. Nonparametric tests are less statistically
powerful than parametric tests.
• 2. The scales of measurement analyzed by
nonparametric tests (i.e., chiefly nominal or
ordinal data) are less sensitive than those
analyzed by parametric tests (i.e., chiefly
interval and ratio data)

Research design Nominal data Ordinal data Parametric Tests
One sample X² goodness-of-fit --- One sample t or z test
Two independent samples X² test of Independ. Mann-Whitney U test Independent groups t test
Two dependent samples --- WilcoXon matched-pairs Dependent groups t test
signed-rank test
More than three Independ X² test of Indep. --- One-way ANOVA
samples
Correlation ----- Spearman rs Pearson r
Nonparametric Tests for
Research Designs and the Nonparametric and Parametric Tests Available to Analyze Their Data

Nonparametric Tests
• Nonparametric test used to analyze nominal
data basically infer whether the pattern
discerned across some number of categories is
anticipated by chance or not.
• When nonparametric tests are applied to
ordinal data, they are often used to discern
whether one sample of observations has
higher rankings than another sample.

The Chi-Square (X²) Test for Categorical
Data
• Categorical data – keeping track of
frequencies
• Frequencies refer to the number of
observations or items than can be
meaningfully grouped under some heading or
label.
• For example, “Are there more women over 6
feet than under that height in a sample"

Chi-Square (X² )
• Chi-Square (X²) compares observed
frequencies against expected frequencies.
• It also makes inference about the presence or
absence of some pattern. , “Are there more
women over 6 feet than under that height in a
sample"

Chi-Square (X² ) Statistical
Assumptions
• 1 Chi-Square (X² ) is not a single distribution but rather a collection
of similar curves, each of which is based on some number of
degrees of freedom.
• 2. Chi-Square (X² ) is similar to the t distribution in that it is based
on one degree of freedom value ( in contrast the F statistic is always
based on two separate degrees of freedom values.)
• 3. Chi-Square (X² ) need not be applied to data that conform to any
particular shape (i.e., normal distribution), though the observations
must be nominal.
• 4. Chi-Square (X² ) observations are randomly selected from some
larger population.
• 5. Chi-Square (X² ) number of expected observations within a given
category should be reasonably large. A good rule of thumb is to
have few (preferably no) categories with less than an expected
frequency of 5 observations in any given analysis.

The Chi-square Test for One-variable:
Goodness-of-Fit
• We work with Nominal data and therefore concentrate
on categories in order to explore the relative
frequencies or proportions present in some
distributions.
• Examples:
• A. How many experimental psychologists are males
rather than females?
• B. Of the four most popular majors at the college,
which one do sophomores choose most often?
• Here FREQUENCIES and COMPARISONS are made
based on how the observations are categorized.
• Statistical test is called Goodness of Fit

Goodness-of-Fit
• Using sample data, the Chi-square test for goodness-of-fit tests
whether obtained observations conform to or "fit" or diverge from
the population proportions specified by a null hypothesis.
• Goodness-of-fit points to the comparison of what pattern or
distribution of frequencies would be anticipated due to chance
versus the one that is actually obtained.
• When the fit between “observed“ and “expected“ observations is
good, then we know that the distribution of observation across the
available categories is more or less equal.
• When the discrepancy between observed and expected
observations is sufficiently large, however, then a significant
difference is likely to be found between or among the categories.
• The departure of observed from expected results means that the fit
is not " good,“ that some influential factors or factor is presumably
causing it.

Example to Illustrate
• A statistics instructor asks the 35 students in
her class to complete a standard course
evaluation. One of the key questions of
interest to this instructor was:
• "Statistic was my favorite class this semester.“

OBSERVED DATA
Strongly agree Agree Undecided Disagree Strongly disagree
17 8 3 2 5
"Statistic was my favorite class this semester.“ (N = 35)
Chi-square test statistic X² indicates whether there is a
difference between some observed set of frequencies – the
data drawn from a piece of research– and a set of expected
frequencies. These expected frequencies constitute the
predictions made under the null hypothesis.

Steps for Testing a Hypothesis Using a Nonparametric Test
1. Verify that the data are based on a Nominal or an ordinal
scale, not on an interval or a ratio scale
2. State the null (H0) and the alternative (H1) hypothesis
3. Select a significance level (p value or 𝛼 level) for the
nonparametric test.
4. Perform the analyses using the statistic, determining
whether to accept or reject (H0). Interpret and evaluate the
results in light of the hypothesis, and if necessary, go to
step 5.
5. Compute any supporting statistics

Chi-Square test for Goodness-of-Fit
• It involves: 2 forms
• No frequency difference among a set of
different categories
• No frequency difference from a comparison
population.

Chi-Square test for Goodness-of-Fit
• Ho: No difference in course ratings across the five rating categories ( i.e.,
strongly agree to strongly disagree )
• Because there are 35 students, the EXPECTED FREQUENCY for each
category when no difference exists would 7 (35 ÷
5 possible categories equals/ = 7 students in each one)

EXPECTED DATA UNDER H0
Strongly agree Agree Undecided Disagree Strongly Disagree
7 7 7 7 7
The general rule of thumb for determining the expected frequencies for the chi-
square test for goodness-of-fit test is simply dividing N by the number of
available categories.
The alternative hypothesis, then, is
H1: There is statistically reliable difference between the observed and
expected frequencies.

COMPARISON DATA UNDER H0
STRONGLY AGREE AGREE UNDECIDED DISAGREE STRONGLY DISAGREE
4 10 11 5 5
H0: No difference in course ratings from prior semester across
the five rating categories (i.e., strongly agree to strongly disagree)
H1: There is a statistically reliable difference between the
observed and the comparison frequencies.

Calculation for Test Statistic : CHI-SQUARE TEST
X² = Σ (fo- fE)²
fE

Calculating X² Test Statistic Using a Tabular Format
fo fE (fo-fE) (fo-fE) ² (fo-fE)²/fE
Strongly agree 17 7
Agree 8 7
Uncertain 3 7
Disagree 2 7
Strongly disagree 5 7
Σfo = 35 ΣfE 35
Note: N = 35 (i.e., Σfo) respondents. These data are based on
hypothetical responses to the statement “Statistics was my
favorite class this semester.“

Chi-square (X²) = 20.86
Dfx = K-1 ( The original rating scale is based
on 5 categories, so the degree of freedom for
this X² test statistic are):
Dfx = 5-1
Dfx = 4.
X² (4, N = 35) = 20.86, p < .05 [APA Style]

Some nonparametric statistic for categorical & ordinal data

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