8 Key excerpts on "Chi Square Test for Independence"

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  eBook - ePub

  Essential Statistics for Public Managers and Policy Analysts

  • Evan M. Berman, XiaoHu Wang(Authors)
  • 2016(Publication Date)
  • CQ Press

    (Publisher)

  When analysts are confronted with two categorical variables, which can also be used to make a contingency table, chi-square is a widely used test for establishing whether a relationship exists (see the Statistics Roadmap at the beginning of the book). Chi-square has three test assumptions: (1) that variables are categorical, (2) that observations are independent, and (3) that no cells have fewer than five expected frequency counts. Remember, violation of test assumptions invalidates any test result. Chi-square is but one statistic for testing a relationship between two categorical variables.

  Once analysts have determined that a statistically significant relationship exists through hypothesis testing, they need to assess the practical relevance of their findings. Remember, large datasets easily allow for findings of statistical significance. Practical relevance deals with the relevance of statistical differences for managers; it addresses whether statistically significant relationships have meaningful policy implications.

  Key Terms

  • Alternate hypothesis (p. 182)
  • Chi-square (p. 178)
  • Chi-square test assumptions (p. 186)
  • Critical value (p. 184)
  • Degrees of freedom (p. 184)
  • Dependent samples (p. 186)
  • Expected frequencies (p. 179)
  • Five steps of hypothesis testing (p. 184)
  • Goodness-of-fit test (p. 191)
  • Independent samples (p.186)
  • Kendall’s tau-c (p.193)
  • Level of statistical significance (p. 183)
  • Null hypothesis (p. 181)
  • Purpose of hypothesis testing (p. 180)
  • Sample size (and hypothesis testing) (p. 188)
  • Statistical power (p. 190)
  • Statistical significance (p. 183)

  Appendix 11.1: Rival Hypotheses: Adding a Control Variable

  We now extend our discussion to rival hypotheses. The following is but one approach (sometimes called the “elaboration paradigm”), and we provide other (and more efficient) approaches in subsequent chapters. First mentioned in Chapter 2 , rival hypotheses are alternative, plausible explanations of findings. We established earlier that men are promoted faster than women, and in Chapter 8

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  eBook - ePub

  Practical Statistics for Field Biology

  • Jim Fowler, Lou Cohen, Philip Jarvis(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  13

  ANALYSING FREQUENCIES

  13.1 The chi-square test

  Field biologists spend a good deal of their time counting and classifying things on nominal scales such as species, colour and habitat. Statistical techniques which analyse frequencies are therefore especially useful. The classical method of analysing frequencies is the chi-square test. This involves computing a test statistic which is compared with a chi-square (χ 2 ) distribution that we outlined in Section 11.11. Because there is a different distribution for every possible number of degrees of freedom (df), tables in Appendix 3 showing the distribution of χ 2 are restricted to the critical values at the significance levels we are interested in. There we give critical values at P = 0.05 and P = 0.01 (the 5% and 1% levels) for 1 to 30 df. Between 30 and 100 df, the critical values are estimated by interpolation, but the need to do this arises infrequently.

  Chi-square tests are variously referred to as tests for homogeneity, randomness, association, independence and goodness of fit. This array is not as alarming as it might seem at first sight. The precise applications will become clear as you study the examples. In each application the underlying principle is the same. The frequencies we observe are compared with those we expect on the basis of some Null Hypothesis. If the discrepancy between observed and expected frequencies is great, then the value of the calculated test statistic will exceed the critical value at the appropriate number of degrees of freedom. We are then obliged to reject the Null Hypothesis in favour of some alternative.

  The mastery of the method lies not so much in the computation of the test statistic itself but in the calculation of the expected frequencies. We have already shown some examples of how expected frequencies are generated. They can be derived from sample data (Example 7.5) or according to a mathematical model (Section 7.4). The versions of the test which compare observed frequencies with those expected from a model are called goodness of fit tests. All versions of the chi-square test assume that samples are random and observations are independent.

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  eBook - ePub

  Sensory Evaluation of Food

  Statistical Methods and Procedures

  • Michael O'Mahony(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  6

  Chi-Square

  6.1 What is Chi-Square?

  We now examine a test called chi-square or chi-squared (also written as χ 2 , where χ is the Greek lowercase letter chi); it is used to test hypotheses about frequency of occurrence. As the binomial test is used to test whether there may be more men or women in the university (a test of frequency of occurrence in the “men” and “women” categories), chi-square may be used for the same purpose. However, chi-square has more uses because it can test hypotheses about frequency of occurrence in more than two categories (e.g., dogs vs. cats vs. cows vs. horses). This is often used for categorizing responses to foods (“like” vs. “indifferent” vs. “dislike” or “too sweet” vs. “correct sweetness” vs. “not sweet enough”).

  Just as there is a normal and a binomial distribution, there is also a chi-square distribution, which can be used to calculate the probability of getting our particular results if the null hypothesis were true (see Section 6.6 ). In practice, a chi-square value is calculated and compared with the largest value that could occur on the null hypothesis (given in tables for various levels of significance); if the calculated value is larger than this value in the tables, H 0 is rejected. This procedure will become clearer with examples.

  In general, chi-square is given by the formula

  Chi-square = Σ [

  ( O − E )

  2

  E

  ]

  where

  O = observed frequency

  E =

  expected  frequency

  We will now examine the application of this formula to various problems. First we look at the single-sample case, where we examine a sample to find out something about the population; this is the case in which a binomial test can also be used.

  6.2 Chi-Square: Single-Sample Test-One-Way Classification

  In the example we used for the binomial test (Section 5.2 ) we were interested in whether there were different numbers of men and women on a university campus. Assume that we took a sample of 22 persons, of whom 16 were male and 6 were female. We use the same logic as with a binomial test. We calculate the probability of getting our result on H 0 , and if it is small, we reject H 0 . From Table G.4.b , the two-tailed binomial probability associated with this is 0.052, so we would not reject H 0 at p < 0.05. However, we can also set up a chi-square test. If H 0 is true, there is no difference in the numbers of men and women; the expected number of males and females from a sample of 22 is 11 each. Thus we have our observed frequencies (O = 16 and 6) and our expected frequencies (E

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  eBook - ePub

  A Conceptual Guide to Statistics Using SPSS

  • Elliot T. Berkman, Steven P. Reise(Authors)
  • 2011(Publication Date)
  • SAGE Publications, Inc

    (Publisher)

  3

  The Chi-Squared Test for Contingency Tables

   

  CHAPTER OUTLINE

  Introduction to the Chi-Squared Test Computing the Chi-Squared Test in SPSS

  A Closer Look: Fisher’s Exact Test

  The Chi-Squared Test for Testing the Distribution of One Categorical Variable

  Introduction to the Chi-Squared Test

  In the chapter on descriptive statistics, we drew a distinction between categorical and continuous variables. Most of the inferential statistics we discuss in this book assume that your outcome variable is continuous. However, sometimes we have outcomes that fall into categories (e.g., was someone on trial for a crime convicted or not? Or did a participant choose to open door number one, two, or three?). In these cases, and when our predictor variable is also categorical, the chi-squared test is appropriate.

  The raw data typically analyzed using the chi-squared test are counts of the same outcome in each of two or more conditions. For example, if you wanted to know whether gender affected traffic court convictions, you could tally up the number of men who were and weren’t convicted on a given day, then separately tally up the number of women who were and weren’t convicted on that same day. Those four counts would then be entered into a chi-squared analysis, and it would tell you whether the proportion of men who were convicted that day was different from the proportion of women who were convicted.

  The chi-squared test can also be used to answer questions about proportions within a single variable. In other words, the test can be used to tell you whether the percentage of cases in each category differs from some hypothesized distribution. Suppose that a court claims that it convicts 90% of people who come up with a traffic violation. If you tallied the number of people who were convicted or not for a given period, the chi-squared test could tell you whether the proportion of people convicted in that period was significantly different from 90%.

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  eBook - ePub

  Statistics for Psychologists

  An Intermediate Course

  • Brian S. Everitt(Author)
  • 2001(Publication Date)
  • Psychology Press

    (Publisher)

  pairs of observations that both have A, and so on.
• To test whether the probability of having A differs in the matched populations, the relevant test statistic is which, if there is no difference, has a chi-squared distribution with a single degree of freedom.
We can illustrate the use of Fisher’s exact test on the data on suicidal feelings in Table 9.4 because this has some small expected values (see Section 9.4 for more comments). The p-value form applying the test is .235, indicating that diagnosis and suicidal feelings are not associated.
To illustrate McNemar’s test, we use the data shown in Table 9.6 . For these data the test statistic takes the value 1.29, which is clearly not significant, and we can conclude that depersonalization is not associated with prognosis where endogenous depressed patients are concerned.
Table 9.6   Recovery of 23 Pairs of Depressed Patients
9.3.  Beyond the Chi-Square Test: Further Exploration of Contingency Tables by Using Residuals and Correspondence Analysis
A statistical significance test is, as implied in Chapter 1 , often a crude and blunt instrument. This is particularly true in the case of the chi-square test for independence in the analysis of contingency tables, and after a significant value of the test statistic is found, it is usually advisable to investigate in more detail why the null hypothesis of independence fails to fit. Here we shall look at two approaches, the first involving suitably chosen residuals and the second that attempts to represent the association in a contingency table graphically.
9.3.1.  The Use of Residuals in the Analysis of Contingency Tables
After a significant chi-squared statistic is found and independence is rejected for a two-dimensional contingency table, it is usually informative to try to identify the cells of the table responsible, or most responsible, for the lack of independence. It might be thought that this can be done relatively simply by looking at the deviations of the observed counts in each cell of the table from the estimated expected values under independence, that is, by examination of the residuals: