Analysis of frequency data: the chisquare test of independence

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Analysis of frequency data the chi-square test
of independence
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Chi-square test of independence

• Is used when
• a score is categorized on two independent
  dimensions
• a between-subjects design is used
• the scores are nominal measures (representing
  only the frequency of occurrence of a response).

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Example

• Is there a statistically significant relationship
  between gender and choice of the program of
  study, - e.g., either physics or library science?
• Would, for example, male students more frequently
  choose the physics program of study than the
  library science program of study?

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Test statistic

• ? chi, the 22nd letter of the Greek alphabet
  (pronounced ky).
• Orc observed frequency in the rc cell, where r
  represents the row number, and c the column
  number.
• Erc expected frequency of the corresponding rc
  cell.
• r number of categories of the row variable.
• c number of levels of the columns variable.
• The double summation signs and limits (
  ) indicate that the
• values of are summed over
  all rows and columns of the
• contingency table.

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Hypothesis testing

• A null hypothesis, H0, and an alternative
  hypothesis, H1, are formulated. The null
  hypothesis provides the sampling distribution of
  the ?2 statistic.
• A significance level a is selected.
• A critical value of ?2, identified as ?2crit, is
  found from the sampling distribution of ?2 that
  can be found in the standard pre-calculated
  statistical tables for the critical values of the
  chi-square distribution.
• Using the value of ?2crit, a rejection region is
  located in the sampling distribution of ?2.
• The observed ?2obs, is calculated from the
  frequencies in the contingency table.
• A decision to reject or not reject H0 is made on
  the basis of whether ?2obs falls into the
  rejection region.

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1) Statistical hypotheses

• H0 The row variable and the column variable are
  independent in the population. In our case
• H0 The program of study choice of the student is
  independent of gender of the student.
• H1 The row and column variables are related in
  the population. In our case
• H1 The program of study choice of the student is
  related to the gender of the student.

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2) Significance level

• In line with common practice, lets choose
• a .05.

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3) Critical value ?2crit

• depends on the degrees of freedom of the
  statistic.
• Degrees of freedom (r - 1)(c 1)
• For our example, r 2 and c 2 thus,
• df (2 - 1)(2 - 1), which equals 1.
• The critical value of ?2 for particular df and a
  can be found in the standard pre-calculated
  statistical tables for the critical values of the
  chi-square distribution in our case ?2crit 3.84

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4) Rejection region

• For our example, if ?2obs is equal to or greater
  than 3.84, it falls into the rejection region and
  will lead to rejection of H0 and acceptance of H1.

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5) Calculating ?2obs
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Practical calculation of ?2obs

• Subtract the expected frequency (E) from the
  observed frequency (O) for each cell.
• Square each O - E difference.
• Divide each squared O - E difference by the
  expected frequency of the cell.
• Sum the resulting (O - E)2/E values over all
  cells in the contingency table to obtain ?2obs.

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Numerical value of ?2obs

• can be calculated as follows

2.286 2.000 1.143 1.000 6.43 when
rounded to two decimal places.
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6) Decisions

• The ?2obs of 6.43 is larger than ?2crit of 3.84
  and, thus, falls into the rejection region.
• H0 is rejected and H1 is accepted.
• There is a statistically significant relationship
  between the row and column variables.
• Gender and program of study selection are related
  in the case of physics and library science.
• A greater than chance frequency of the physics
  program of study selection is associated with
  male gender, and a greater than chance frequency
  of the library science program of study selection
  is associated with female gender.

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Assumptions for the use of the chi-square test of
independence

• The test may be used on a contingency table of
  any size (e.g., 2x3, 3x4, or 4x4).
• Each participant may contribute only one response
  to the contingency table.
• The number of responses obtained should be large
  enough so that no expected frequency is less than
  10 in a 2x2 contingency table, or less than 5 in
  a contingency table larger than 2x2.
• Alternative Fisher exact test.