ChiSquare Test
CHISQUARE TEST
Studies often collect data on categorical variables that can be summarized as a series of counts. These counts are commonly arranged in a tabular format known as a contingency table. For example, a study designed to determine whether or not there is an association between cigarette smoking and asthma might collect data that could be assembled into a 2−2 table. In this case, the two columns could be defined by whether the subject smoked or not, while the rows could represent whether or not the subject experienced symptoms of asthma. The cells of the table would contain the number of observations or patients as defined by these two variables.
The chisquare test statistic can be used to evaluate whether there is an association between the rows and columns in a contingency table. More specifically, this statistic can be used to determine whether there is any difference between the study groups in the proportions of the risk factor of interest. Returning to our example, the chisquare statistic could be used to test whether the proportion of individuals who smoke differs by asthmatic status.
The chisquare test statistic is designed to test the null hypothesis that there is no association between the rows and columns of a contingency table. This statistic is calculated by first obtaining for each cell in the table, the expected number of
Table 1
Observed values for data presented in a twobytwo table  
source: Courtesy of author.  
Variable 2  Variable 1  Total  
Yes  No  
Yes  a  b  a+b 
No  c  d  c+d 
Total  a+c  b+d  n 
events that will occur if the null hypothesis is true. When the observed number of events deviates significantly from the expected counts, then it is unlikely that the null hypothesis is true, and it is likely that there is a rowcolumn association. Conversely, a small chisquare value indicates that the observed values are similar to the expected values leading us to conclude that the null hypothesis is plausible. The general formula used to calculate the chisquare (X ^{2}) test statistic is as follows:
where O = observed count in category; E = expected count in the category under the null hypothesis; df = degrees of freedom; and c, r represent the number of columns and rows in the contingency table.
The value of the chisquare statistic cannot be negative and can assume values from zero to infinity. The pvalue for this test statistic is based on the chisquare probability distribution and is generally extracted from published tables or estimated using computer software programs. The pvalue represents the probability that the chisquare test statistic is as extreme as or more extreme than observed if the null hypothesis were true. As with the t and F distributions, there is a different chisquare distribution for each possible value of degrees of freedom. Chisquare distributions with a small number of degrees of freedom are highly skewed; however, this skewness is attenuated as the number of degrees of freedom increases. In general, the degrees of freedom for tests of hypothesis that involve an r×c contingency table is
Table 2
Expected values for data presented in a twobytwo table  
source: Courtesy of author.  
Variable 2  Variable 1  Total  
Yes  No  
Yes  (a+b)(a+c)/n  (a+b)(b+d)/n  a+b 
No  (c+d)(a+c)/n  (c+d)(b+d)/n  c+d 
Total  a+c  b+d  n 
equal to (r7minus;1)×(c−1); thus for any 2×2 table, the degrees of freedom is equal to one. A chisquare distribution with one degree of freedom is equal to the square root of the normal distribution, and, consequently, either the chisquare or standard normal table can be used to determine the corresponding pvalue.
The chisquare test is most widely used to conduct tests of hypothesis that involve data that can be presented in a 2×2 table. Indeed, this tabular format is a feature of the casecontrol study design that is commonly used in public health research. Within this contingency table, we could denote the observed counts as shown in Table 1. Under the null hypothesis of no association between the two variables, the expected number in each cell under the null hypothesis is calculated from the observed values using the formula outlined in Table 2.
The use of the chisquare test can be illustrated by using hypothetical data from a study investigating the association between smoking and asthma among adults observed in a community health clinic. The results obtained from classifying 150 individuals are shown in Table 3. As Table 3 shows, among asthmatics the proportion of smokers was 40 percent (20/50), while the corresponding proportion among asymptomatic individuals was 22 percent (22/100). By applying the formula presented in Table 2, for the observed cell counts of 20, 30, 22, and 78 (Table 3) the corresponding expected counts are 14, 36, 28, and 72. The observed and expected counts can then be used to calculate the chisquare test statistic as outlined in Equation 1. The resulting value of the chisquare
Table 3
Hypothetical data showing chisquare test  
source: Courtesy of author.  
Symptoms of asthma  Ever smoke cigarettes  Total  
Yes  No  
Yes  20  30  50 
No  22  30  100 
Total  42  108  150 
test statistic is approximately 5.36, and the associated pvalue for this chisquare distribution that has one degree of freedom is 0.02. Therefore, if there was truly no association between smoking and asthma, there is a 2 out of 100 probability of observing a difference in proportions that is at least as large as 18 percent (40%–22%) by chance alone. We would therefore conclude that the observed difference in the proportions is unlikely to be explained by chance alone, and consider this result statistically significant.
Because the construction of the chisquare test makes use of discrete data to estimate a continuous distribution, some authors will apply a continuity correction when calculating this statistic. Specifically,
where ^{O}_{i}−E_{i} is the absolute value of the difference between O_{i} and E_{i} and the term 0.5 in the numerator is often referred to as Yates correction factor. This correction factor serves to reduce the chisquare value, and, therefore, increases the resulting pvalue. It has been suggested that this correction yields an overly conservative test that may fail to reject a false null hypothesis. However, as long as the sample size is large, the effect of the correction factor is negligible.
When there is a small number of counts in the table, the use of the chisquare test statistic may not be appropriate. Specifically, it has been recommended that this test not be used if any cell in the table has an expected count of less than one, or if 20 percent of the cells have an expected count that is greater than five. Under this scenario, the Fisher's exact test is recommended for conducting tests of hypothesis.
Paul J. Villeneuve
(see also: Normal Distributions; Probability Model; Sampling; Statistics for Public Health; TTest )
Bibliography
Cohran, W. G. (1954). "Some Methods for Strengthening the Common X ^{2} Test." Biometrics 10:417–451.
Grizzle, J. E. (1967). "Continuity Correction in the X2 Test for 2×2 Tables." The American Statistician 21:28–32.
Pagano, M., and Gauvreau, K. (2000). Principles of Biostatistics, 2nd edition. Pacific Grove, CA: Duxbury Press.
Rosner, B. (2000). Fundamentals of Biostatistics, 5th edition. Pacific Grove, CA: Duxbury Press.
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ChiSquare
ChiSquare
X ^{2} TEST FOR POPULATION VARIANCES
X ^{2} TESTS OF GOODNESS OF FIT AND INDEPENDENCE
The term chisquare (χ ^{2}) refers to a distribution, a variable that is χ ^{2} distributed, or a statistical test employing the χ ^{2} distribution. A χ ^{2} distribution with k degrees of freedom (df ) has mean k, variance 2k, and mode k – 2 (if k > 2), and is denoted . Much of its usefulness in statistical inference derives from the fact that the sample variance of a normally distributed variable is χ ^{2} distributed with df = N – 1. All χ ^{2} distributions are asymmetrical, rightskewed, and nonnegative. Owing to the broad utility of the χ ^{2} distribution, tabled χ ^{2} probability values can be found in virtually every introductory statistics text.
X ^{2} TEST FOR POPULATION VARIANCES
A test of the null hypothesis that (e.g., H _{0}: σ^{2} 1.8) is conducted by obtaining the sample variance s ^{2}, computing the test statistic
and consulting values of the distribution. For a twotailed test, G is compared to the critical values associated with the lower and upper (50 × α)% of the distribution. Rejection implies, with confidence 1 – α, that the sample is not drawn from a normally distributed population with variance .
X ^{2} TESTS OF GOODNESS OF FIT AND INDEPENDENCE
The χ ^{2} goodness of fit test compares two finite frequency distributions—one a set of observed frequency counts in C categories, the other a set of counts expected on the basis of theory or chance. The statistic
is computed, where O_{i} and E_{i} are, respectively, the observed and expected frequencies for category i given a fixed total sample size N. G is approximately χ ^{2} distributed with df = C – 1. If the null hypothesis of equality is rejected, the test implies a statistically significant departure from expectations.
This test can be extended to test the null hypothesis that several frequency distributions are independent. For example, given a 3 × 4 contingency table of frequencies, where R = 3 rows (conditions) and C = 4 columns (categories), G may be computed as
and compared against a distribution. Expected frequencies are computed as the product of the marginal totals for column i and row j divided by N. Rejection of the null hypothesis implies that not all rows (or columns) were sampled from independent populations. This test may be extended to any number of dimensions.
These χ ^{2} tests have been found to work well with average expected frequencies as low as 2. However, these tests are inappropriate if the assumption of independent observations is violated.
COMPARISON OF DISTRIBUTIONS
A common application of χ ^{2} is to test the hypothesis that a sample’s parent population follows a particular continuous probability density function. The test is conducted by first dividing the hypothetical distribution into C “bins” of equal wdth. The frequencies expected for each bin (E_{i} ) are approximated by computing the probability of randomly selecting a case from that bin and multiplying by N. Observed frequencies (O_{i} ) are obtained by using the same bin limits in the observed distribution. The onetailed test is conducted by using equation 2 and comparing the result to the critical value drawn from a distribution. Note that the number of bins, and points of division between bins, must be chosen arbitrarily, yet these decisions can have a large impact on conclusions.
The χ ^{2} distribution has many other applications in the social sciences, including Bartlett’s test of homogeneity of variance, Friedman’s test for median differences, tests for heteroscedasticity, nonparametric measures of association, and likelihood ratios. In addition, χ ^{2} statistics form the basis for many model fit and selection indices used in latent variable analyses, item response theory, logistic regression, and other advanced techniques. All of these methods involve the evaluation of the discrepancy between a model’s implications and observed data.
SEE ALSO Distribution, Normal
BIBLIOGRAPHY
Howell, David C. 2006. Statistical Methods for Psychology. 6th ed. Belmont, CA: Wadsworth Publishing.
Pearson, Karl. 1900. On the Criterion That a Given System of Deviations From the Probable in the Case of a Correlated System of Variables Is Such That It Can Be Reasonably Supposed To Have Arisen From Random Sampling. Philosophical Magazine 50: 157175.
Kristopher J. Preacher
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chisquared test
chisquared test (χ2) In statistics, a hypothesis test used to determine the goodness of fit of a particular data set with that expected from a theoretical distribution. The test statistic is a function of the difference between observed and expected values which is compared to the chisquared distribution. The chisquared distribution is a distribution of sample variance based on a single parameter, the degrees of freedom.
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chisquare test
chisquare test (kyskwair) n. (in statistics) a test to determine if the difference between two groups of observations is statistically significant (see significance), used in controlled trials and other studies. It measures the differences between theoretical and observed frequencies (see frequency distribution) and identifies whether or not variables are dependent (see variable).
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chisquared test
chisquared test (χ^{2} test) A statistical test that is used to determine whether data obtained by sampling agree with those predicted hypothetically, and thus to test the validity of the hypothesis.
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chisquared test
chisquared test(χ^{2} test) A statistical test that is used to determine whether data obtained by sampling agree with those predicted hypothetically, and thus to test the validity of the hypothesis.
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chisquared test
chisquared test (χ^{2}) A statistical test that is used to determine whether data obtained by sampling agree with those predicted hypothetically, and thus to test the validity of the hypothesis.
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chisquare test
chisquare test: see statistics.
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chisquare
chisquare See SIGNIFICANCE TESTS.
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ChiSquare Test
ChiSquare Test
The chisquare test (X^{2}) is the most commonly used method for comparing frequencies or proportions. It is a statistical test used to determine if
Table 1 . (Thomson Gale.)  

Congenital heart defects in Down and Patau syndrome patients  
Karyotype  
Down syndrome  Patau syndrome  Total  
CHD present  24  20  44 
CHD absent  36  5  41 
Total  60  25  85 
Table 2. 2 × 2 Table summarizing data collected from two groups of patients . (Thomson Gale.)  

Summary of subjects (Down and Patau syndrome patients)  
Karyotype  
Down syndrome  Patau syndrome  Total  
CHD present  O_{11}  O_{12}  r1 
CHD absent  O_{21}  O_{22}  r1 
Total  c1  c2  N 
Table 3. Observed and expected frequencies and X ^{2} for data in Table 1 . (Thomson Gale.)  

Observed and expected frequencies of Down and Patau among subjects  
Observed (o)  Expected (e)  o–e  (o–e)^{2}  (o–e)^{2}/e 
24  31.1  −7.1  50.41  1.62 
20  12.9  7.1  50.41  3.91 
36  28.9  7.1  50.41  1.74 
5  12.1  27.1  50.41  4.17 
85  85.0  x^{2} =11.44 
observed data deviate from those expected under a particular hypothesis. The chisquare test is also referred to as a test of a measure of fit or “goodness of fit” between data.
Typically, the chisquare test examines whether or not two samples are different enough in a particular characteristic to be considered separate from each other. Chisquare analysis belongs to a type of analysis
known as univariate analysis; this analysis examines the possible effect of one variable (often called the independent variable) upon an outcome (often called the dependent variable).
The chisquare analysis is used to test what is termed the null hypothesis (H_{0} ). A null hypothesis states there is no significant difference between expected and observed data. Investigators either accept or reject H_{0} after comparing the value of chisquare to a probability distribution. Chisquare values with low probability lead to the rejection of H_{0}; put another way, this means that a factor other than chance created a large difference between the expected and observed results. Values with a higher probability are accepted; in other words there is no appreciable difference between the observed and expected values.
Chisquare tests only evaluate a single variable, thus they do not take into account the interaction among more than one variable upon the outcome. Therefore, other unseen factors may make the variables appear to be associated even when they are not. Despite this possibility, if properly used, the chisquare test is a very useful tool for the evaluation of associations and can be used as a preliminary analysis of more complex statistical evaluations.
Resources
BOOKS
Campbell, M.J. Statistics at Square Two: Understanding Modern Statistical Applications in Medicine. London: BMJ Publishing Group, 2001.
Donnelly, Robert A., Jr. The Complete Idiot’s Guide to Statistics. New York: Penguin, 2004.
Hunter, G. Scott. Let’s Review BiologyThe Living Environment. Hauppauge NY: Barron’s Educational Series,2003.
Antonio Farina
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ChiSquare Test
Chisquare test
The chisquare test (KHGR2) is the most commonly used method for comparing frequencies or proportions. It is a statistical test used to determine if observed data deviate from those expected under a particular hypothesis. The chisquare test is also referred to as a test of a measure of fit or "goodness of fit" between data. Typically, the hypothesis tested is whether or not two samples are different enough in a particular characteristic to be considered members of different populations. Chisquare analysis belongs to the family of univariate analysis, i.e., those tests that evaluate the possible effect of one variable (often called the independent variable) upon an outcome (often called the dependent variable).
The chisquare analysis is used to test the null hypothesis (H0), which is the hypothesis that states there is no significant difference between expected and observed data. Investigators either accept or reject H0, after comparing the value of chisquare to a probability distribution. Chisquare values with low probability lead to the rejection of H0 and it is assumed that a factor other than chance creates a large deviation between expected and observed results. As with all nonparametric tests (that do not require normal distribution curves), chisquare tests only evaluate a single variable, thus they do not take into account the interaction among more than one variable upon the outcome.
A chisquare analysis is best illustrated using an example in which data from a population is categorized with respect to two qualitative variables. Table 1 shows a sample of patients categorized with respect to two qualitative variables, namely, congenital heart defect (CHD; present or absent) and karyotype (trisomy 21, also called Down syndrome , or trisomy 13, also called Patau syndrome ). The classification table used in a chisquare analysis is called a contingency table and this is its simplest form (2 x 2). The data in a contingency table are often defined as row (r) and column (c) variables.
In general, a chisquare analysis evaluates whether or not variables within a contingency table are independent, or that there is no association between them. In this example, independence would mean that the proportion of individuals affected by CHD is not dependent on karyotype; thus, the proportion of patients with CHD would be similar for both Down and Patau syndrome patients. Dependence, or association, would mean that the proportion of individuals affected by CHD is dependent on kayotype, so that CHD would be more commonly found in patients with one of the two karyotypes examined.
Table 1 shows a 2 x 2 contingency table for a chisquare test—CHD (congenital heart defects) found in patients with Down and Patau syndromes
Chisquare is the sum of the squared difference between observed and expected data, divided by the expected data in all possible categories:
Χ2 = (O11  E11)2 / E11 + (O12  E12)2 / E 12 + (O21  E21)2/ E21 + (O22–e22)2 / E22, where O11 represents the observed number of subjects in column 1, row 1, and so on. A summary is shown in Table 2.
The observed frequency is simply the actual number of observations in a cell. In other words, O11 for CHD in the Downsyndromeaffected individuals is 24. Likewise, the observed frequency of CHD in the Patausyndromeaffected patients is 20 (O12). Because the null hypothesis assumes that the two variables are independent
Karyotype  
Down syndrome  Patau syndrome  Total  
Congenital Heart Defects  CHD present  24  20  44 
CHD absent  36  5  41  
Total  60  25  85 
Karyotype  
Down syndrome  Patau syndrome  Total  
Congenital Heart Defects  CHD present  O11  O12  r1 
CHD absent  O21  O22  r2  
Total  c1  c2  N 
Observed (o)  Expected (e)  o e  (o e) 2  (o e) 2/e 
24  31.1  7.1  50.41  1.62 
20  12.9  7.1  50.41  3.91 
36  28.9  7.1  50.41  1.74 
5  12.1  7.1  50.41  4.17 
85  85.0  Χ2=11.44 
of each other, expected frequencies are calculated using the multiplication rule of probability. The multiplication rule says that the probability of the occurrence of two independent events X and Y is the product of the individual probabilities of X and Y. In this case, the expected probability that a patient has both Down syndrome and CHD is the product of the probability that a patient has Down syndrome (60/85 = 0.706) and the probability that a patient has CHD (44/85 = 0.518), or 0.706 x 0.518 = 0.366. The expected frequency of patients with both Down syndrome and CHD is the product of the expected probability and the total population studied, or 0.366 x 85 = 31.1.
Table 3 presents observed and expected frequencies and Χ2 for data in Table 1.
Before the chisquare value can be evaluated, the degrees of freedom for the data set must be determined. Degrees of freedom are the number of independent variables in the data set. In a contingency table, the degrees of freedom are calculated as the product of the number of rows minus 1 and the number of columns minus 1, or (r1)(c1). In this example, (21)(21) = 1; thus, there is just one degree of freedom.
Once the degrees of freedom are determined, the value of Χ2 is compared with the appropriate chisquare distribution, which can be found in tables in most statistical analyses texts. A relative standard serves as the basis for accepting or rejecting the hypothesis. In biological research, the relative standard is usually p = 0.05, where p is the probability that the deviation of the observed frequencies from the expected frequencies is due to chance alone. If p is less than or equal to 0.05, then the null hypothesis is rejected and the data are not independent of each other. For one degree of freedom, the critical value associated with p = 0.05 for Χ2 is 3.84. Chisquare values higher than this critical value are associated with a statistically low probability that H0 is true. Because the chisquare value is 11.44, much greater than 3.84, the hypothesis that the proportion of trisomy13affected patients with CHD does not differ significantly from the corresponding proportion for trisomy21affected patients is rejected. Instead, it is very likely that there is a dependence of CHD on karyotype.
Figure 1 shows chisquare distributions for 1, 3, and 5 degrees of freedom. The shaded region in each of the distributions indicates the upper 5% of the distribution. The critical value associated with p = 0.05 is indicated. Notice that as the degrees of freedom increases, the chisquare value required to reject the null hypothesis increases.
Because a chisquare test is a univariate test; it does not consider relationships among multiple variables at the same time. Therefore, dependencies detected by chisquare analyses may be unrealistic or noncausal. There may be other unseen factors that make the variables appear to be associated. However, if properly used, the test is a very useful tool for the evaluation of associations and can be used as a preliminary analysis of more complex statistical evaluations.
See also Bioinformatics and computational biology; Statistics.
Resources
books
Grant, Gregory R., and Warren J. Ewens. Statistical Methods inBioinformatics. New York: Springer Verlag, 2001.
Nikulin, Mikhail S., and Priscilla E. Greenwood. A Guide toChiSquare Testing. New York: WileyInterscience, 1996.
other
Rice University. Rice Virtual Lab in Statistics. "Chi Square Test of Deviations." (cited November 20, 2002). <http;://www. ruf.rice.edu/~lane/stat_sim/chisq_theor/>.
Antonio Farina
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"ChiSquare Test." The Gale Encyclopedia of Science. . Encyclopedia.com. (September 11, 2018). http://www.encyclopedia.com/science/encyclopediasalmanacstranscriptsandmaps/chisquaretest0
"ChiSquare Test." The Gale Encyclopedia of Science. . Retrieved September 11, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/encyclopediasalmanacstranscriptsandmaps/chisquaretest0
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