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Chi-square statistic

- Overview
- Two types of statistics parametric and

non-parametric - Parametric statistics are statistics that measure

parameters of underlying distributions e.g., ?

and ? of a normal distribution - Distribution free or non-parametric statistics

are statistics that make few assumptions about

the underlying population distribution

Chi-square statistic

- Overview
- This lecture will introduce the chi square

statistic, a statistic that is used to analyze

and interpret frequency data - Note unlike t tests and z scores, it is not

assumed that there is an underlying numerical

score ? - Each person contributes exactly one observation

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Chi-square test for goodness of fit uses sample

data to test hypotheses about the shape or

proportions of a population distribution. The

test determines how well the obtained sample

proportions fit the population proportions

specified by the null hypothesis

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Null hypothesis for chi square test states the

proportion of the population in each category.

This proportion is determined by the question

that the investigator wishes to address

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Typical questions are that there is no preference

across categories - E.g., brand x, brand y, brand z are equally

preferred or preferred by 1/3 of the sample - Sample does not differ from comparison

population. - E.g., Canadian sample does not differ from an

American sample - E.g., distribution does not differ from

theoretically obtained distribution of responses

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- data consist of observations from n individuals,

each of whom contributes a single observation,

and is classified into one category - Categories are mutually exclusive and exhaustive
- E.g., suppose 40 seniors are identified as at

risk for stroke or not at risk 10 of the seniors

are at risk and 30 are not at risk

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Observed frequency number of individuals from a

sample that fall into a particular category - Note each individual is counted in one and only

one category in other words the categories are

mutually exclusive and exhaustive

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Expected frequency
- In contrast to observed frequency, expected

frequency refers to the expected frequency as

determined from the null hypothesis - E.g., suppose the expected frequency, based on a

national survey, was that 60 of the seniors

should be at risk then in a sample of 40 seniors

one would expect that 24 should be at risk (since

.6 40 24), and 16 will not be at risk

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Expected frequency frequency predicted from

null hypothesis and sample size (n). - It can be determined by multiplying the

proportion expected from the null hypothesis

times the sample size n - fe p n, where fe is the expected frequency, p

is the proportion, and n is the sample size

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- chi-square X2 S (fo fe)2 / fe ,
- Where fo , fe are observed and expected

frequencies - Note when the observed and expected frequencies

are similar the chi square value should be small

this will occur when the null hypothesis is true

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- chi-square X2 S (fo fe)2 / fe ,
- Critical region found when chi square values are

large - Chi square values larger when the number of

categories is larger - different chi square distribution for different

degrees of freedom (df) - df C 1, where C number of categories

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- chi-square X2 S (fo fe)2 / fe ,
- null hypothesis is rejected when the obtained

chi-square value is greater than that expected

for a given alpha level, see Table B.8 - E.g., suppose alpha .05 and df 5, recall df

C 1, H0 is rejected if X2 11.07

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Applying chi-square statistic
- Example.
- Purpose to determine factors involved in course

selection. Students must select one of the

following alternatives - Interest in course topic
- Ease of passing course
- Instructor for course
- Time of day of course

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Applying chi-square statistic
- Example.
- State hypothesis
- H0 there is no preference for the four factors.

Hence probability of selecting each factor is ¼ - H1 the factors differ in preference
- Set alpha level at .05
- df C 1 3
- Critical region chi-square 7.81

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Applying chi-square statistic
- Example.
- Determine expected frequency as shown in next

table - Then compute X2

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- X2 S (fo fe)2 / fe ,
- (18 12.5)2/12.5 (17 12.5)2/12.5 (7

12.5)2/12.5 (8 12.5)2/12.5 - 8.08
- Make decision
- Reject H0 because chi square value is in critical

region

Chi-square statistic

- Chi-square (X2) test for goodness of fit
- Reporting result APA style
- As shown in Table X, students were more likely to

endorse some factors than others in course

selection. These differences in preference were

statistically significant, X2 (3, n 50) 8.08,

p lt .05.

Chi-square statistic

- Chi-square (X2) test for independence
- Chi-square statistic may also be used to test

whether there is a relation between two variables - Here individual is measured on two variables that

are categorical in nature - Data are presented in form of a table

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for independence
- Is there a relation between personality and

colour preference? - What makes it hard to tell is that there are

differences in sample size of the introverts and

extroverts - 1 approach convert to proportions proportions

should be similar

Chi-square statistic

- Chi-square (X2) test for independence
- Calculation involves comparing observed to

expected frequencies

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for independence
- how would you describe in words the primary

finding in this table?

Chi-square statistic

- Chi-square (X2) test for independence
- To calculate the expected frequency of a given

cell, multiply the marginal row (fr ) and column

(fc ) frequencies, and then divide by the total

number of observations - fe fc fr / n

Chi-square statistic

Chi-square statistic

- Chi-square (X2) test for independence
- chi-square X2 S (fo fe)2 / fe
- df (C-1) (R-1), where C column, R row
- In a 2 X 4 matrix this means that there are 3 df

because entering 3 values in the matrix allows

one to fill in the remainder of the matrix, given

knowledge of the marginal values

Chi-square statistic

Chi-square statistic

- Choosing the right test
- Suppose you want to investigate the relationship

between self-esteem and academic children - The statistical test one employs depends upon the

experimental design and the nature of the

measures used

Chi-square statistic

- Choosing the right test
- Pearson correlation can be used if the two

measures are interval or ratio, and each person

is measured on both - Chi square can be used if two measures are

categorized into a small number of categories,

and each person is classified into one of the

categories (e.g., academic performance, and self

esteem divided into 3 categories)

Chi-square statistic

- Choosing the right test
- Independent measures t-test can be used if

participants are divided into two groups (e.g.,

high, low self esteem) and academic performance

is measured

Chi-square statistic

- Chi-square test of independence
- Example.
- Want to investigate relation between academic

performance and self esteem. Test n 150

children on self esteem and academic performance

Chi-square statistic

Chi-square statistic

- Chi-square test of independence
- State hypothesis (2 equivalent alternatives)
- H0 there is no relationship between self esteem

and academic performance (like correlation) - H0 the distribution of self esteem does not

differ between low and high academic performers

(like t-test)

Chi-square statistic

- Chi-square test of independence
- Calculate df
- df (R-1) (C-1) (2-1) (3-1) 2
- Determine critical value
- df 2, alpha .05 X2 5.99

Chi-square statistic

- Determine expected frequencies

Chi-square statistic

Chi-square statistic

Chi-square statistic

- Compute X2
- chi-square X2 S (fo fe)2 / fe
- (17 12)2/12 (32 30)2/ 30
- (34 27)2/27
- 8.22
- Make decision
- X2 (2, n 150) 8.22, p lt .05, why?

Chi-square statistic

- Chi-square test assumptions
- Independence of observations
- Expected frequencies
- Should not be less than 5

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