Chisquare statistic

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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

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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

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Chi-square statistic
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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

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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

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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

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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

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Chi-square statistic
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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

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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

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Chi-square statistic
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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

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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

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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

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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

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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

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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

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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

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Chi-square statistic
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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

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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.

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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

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Chi-square statistic
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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

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Chi-square statistic

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

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Chi-square statistic
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Chi-square statistic

• Chi-square (X2) test for independence
• how would you describe in words the primary
  finding in this table?

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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

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Chi-square statistic
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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

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Chi-square statistic
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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

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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)

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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

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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

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Chi-square statistic
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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)

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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

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Chi-square statistic

• Determine expected frequencies

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Chi-square statistic
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Chi-square statistic
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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?

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Chi-square statistic

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