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Chapter 11 Categorical Data Analysis Categorical Data and the - - PowerPoint PPT Presentation
Chapter 11 Categorical Data Analysis Categorical Data and the - - PowerPoint PPT Presentation
Chapter 11 Categorical Data Analysis Categorical Data and the Multinomial Distribution Properties of the Multinomial Experiment 1. Experiment has n identical trials 2. There are k possible outcomes to each trial, called classes, categories or
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Testing Category Probabilities: One-Way Table
In a multinomial experiment with categorical data from a single qualitative variable, we summarize data in a one-way table.
Schema for one-way table for an experiment with k outcomes
k1 k2 … k n1 n2 … nk
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Testing Category Probabilities: One-Way Table
Hypothesis Testing for a One-Way Table
- Based on the 2 statistic, which allows comparison
between the observed distribution of counts and an expected distribution of counts across the k classes
- Expected distribution = E(nk)=npk, where n is the total
number of trials, and pk is the hypothesized probability of being in class k according to H0
- The test statistic, 2, is calculated as
and the rejection region is determined by the 2 distribution using k-1 df and the desired
2 2 1
( )
k i i i i
n E n E n
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Testing Category Probabilities: One-Way Table
Hypothesis Testing for a One-Way Table
- The null hypothesis is often formulated as a no difference,
where H0: p1=p2=p3=…=pk=1/k, but can be formulated with non-equivalent probabilities
- Alternate hypothesis states that Ha: at least one of the
multinomial probabilities does not equal its hypothesized value
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Testing Category Probabilities: One-Way Table
Hypothesis Testing for a One-Way Table
- The null hypothesis is often formulated as a no
difference, where H0: p1=p2=p3=…=pk=1/k, but can be formulated with non-equivalent probabilities
- Alternate hypothesis states that Ha: at least one of
the multinomial probabilities does not equal its hypothesized value
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Testing Category Probabilities: One-Way Table
One-Way Tables: an example
H0: pnone=.10, pStandard=.65, pMerit=.25 Ha: At least 2 proportions differ from proposed plan Rejection region with =.01, df = k-1 = 2 is 9.21034 Since the test statistic falls in the rejection region, we reject H0
=Total x p
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Testing Category Probabilities: One-Way Table
Conditions Required for a valid 2 Test
- Multinomial experiment has been
conducted
- Sample size is large, with E(ni) at least 5 for
every cell
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Testing Category Probabilities: Two-Way (Contingency) Table
Used when classifying with two qualitative variables H0: The two classifications are independent Ha: The two classifications are dependent Test Statistic: Rejection region:2>2
, where 2 has (r-1)(c-1) df
General r x c Contingency Table Column 1 2 … c Row Totals 1 n11 n12 … n1c R1 2 n21 n22 n2c R2 Row … … … … … … r nr1 nr2 … nrc Rr Column Totals C1 C2 … Cc n
2 2 ij ij i j ij ij
n E R C w h e re E E n
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Testing Category Probabilities:
Two-Way (Contingency) Table
Conditions Required for a valid 2 Test
- N observed counts are a random sample
from the population of interest
- Sample size is large, with E(ni) at least 5 for
every cell
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Testing Category Probabilities:
Two-Way (Contingency) Table
Sample Statistical package output
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A Word of Caution about Chi-Square Tests
- When an expected cell count is less than 5,
2 probability distribution should not be used
- If H0 is not rejected, do not accept H0 that
the classifications are independent, due to the implications of a Type II error.
- Do not infer causality when H0 is rejected.