Chi-square test Testing for independeny The r x c contingency tables square test

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1 Chi-square test Testing for independeny The r x c contingency tables square test 1

2 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach

3 Example A study was carried out to investigate the proportion of persons getting influenza vary according to the type of vaccine. Given below is a 3 x table of observed frequencies showing the number of persons who did or did not get influenza after inoculation with one of three vaccines. Does proportion of getting influenza depend on the type of vaccine? Type of vaccine Number getting influenza Number not getting influenza Total Seasonal only 43 (15.35%) (100%) H1N1 only 5 (0.8%) (100%) Combined 5 (9.%) (100%) Totals HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 3

4 Test of independence In biology the most common application for chi-squared is in comparing observed counts of particular cases to the expected counts. A total of n experiments may have been performed whose results are characterized by the values of two random variable X and Y. We assume that the variables are discrete and the values of X and Y are x 1, x,...,x r and y 1, y,...,y c, respectively, which are the outcomes of the events A 1,A,...,A r and B 1, B,...,B c. Let's denote by k ij the number of the outcomes of the event (A i, B j ). These numbers can be grouped into a matrix, called a contingency table. It has the following form: HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 4

5 Contingency table B 1 B... B c Total A 1 k 11 k 1... k 1c k 1 + A k 1 k... k c k Frequency of A i event i1,, r c k i + k ij j 1 A r k r1 k r... k rc k r + Total k + 1 k +... k +c n Frequency of B i event j1,, c k r + j k ij i 1 HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 5

6 Chi-square test (Pearson) H 0 : The two variables are independent. Mathematically: P(A i B j ) P (A i ) P( B j ) Test statistic: χ r s ( k ij k i 1 j 1 + If H 0 is true, then χ has asymptotically χ distribution with (r-1)(c-1) degrees of freedom., namely (number of rows -1)(number of columns - ) Decision: if χ > χ table then we reject the null hypothesis that the two variables are independent, in the k + i + n k n k j + j ) opposite case we do not reject the null hypothesis. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 6

7 Observed and expected frequencies B 1 B... B j... B c Total A 1 k 11 k 1... k 1j... k 1c k 1 + A k 1 k... k j... k c k A i k i1 k i... k ij... k ic k i A r k r1 k r... k rj... k rc k r + χ r s r i 1 j 1 s ij ij ij ij i + i 1 j 1 i + + j ( O ( k E E k k n ) n k k + j ) Total k + 1 k +... k + j... k + c n Observed (O ij ) k ij Expected (E ij ): Row total*column total/n HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 7 k i + k+ n j

8 Example A study was carried out to investigate the proportion of persons getting influenza vary according to the type of vaccine. Given below is a 3 x table of observed frequencies showing the number of persons who did or did not get influenza after inoculation with one of three vaccines. Type of vaccine Number getting influenza Number not getting influenza Total Seasonal only H1N1 only Combined Totals There are two categorical variables (type of vaccine, getting influenza) H 0 : The two variables are independent proportions getting influenza are the same for each vaccine HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 8

9 Type of vaccine Calculation of the test statistic Observed frequencies Number getting influenza Number not getting influenza Tot al Type of vaccine Expected frequencies Number getting influenza Number not getting influenza Tot al Seasonal only H1N1 only Combined Totals Seasonal only H1N1 only Combined Totals χ ki+ k+ j ( kij ) n (43 4) (37 38) (5 37.5) ( ) (5 40.5) (45 9.5) ki + k j n r s i 1 j 1 + χ² χ Degrees of freedom: {(r 1 )(c 1 )} (-1)*(3-1) Here χ > χ table 5.991; (df;α0.05). We reject the null hypothesis We conclude that the proportions getting influenza are not the same for each type of vaccine HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 9

10 Assumption of the chi-square test Expected frequencies should be big enough The number of cells with expected frequencies < 5 can be maximum 0% of the cells. For example, in case of 6 cells, expected frequencies <5 can be in maximum 1 cell (0% of 6 is 1.) HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 10

11 SPSS results Chi-Square Tests Pearson Chi-Square Likelihood Ratio Linear-by-Linear Association N of Valid Cases a. χ² and p0.001 Asymp. Sig. Value df (-sided) 13,603 a,001 13,941,001 3,878 1, cells (,0%) have expected count less than 5. The minimum expected count is 37,50. Here p0.001 < α0.05 we reject the null hypothesis. We conclude that the proportions getting influenza are not the same for each type of vaccine HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 11

12 The chi-square test for goodness of fit Goodness of fit tests are used to determine whether sample observation fall into categories in the way they "should" according to some ideal model. When they come out as expected, we say that the data fit the model. The chi-square statistic helps us to decide whether the fit of the data to the model is good. H0: the distribution of the variable X is a given distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 1

13 The distribution of the sample depending on the type of the variable Categorical variable. Example. A dice is thrown 10 times. We would like to test whether the dice is fair or biased. Observed frequencies Continuous variable Example. We would like to test whether the sample is drawn from a normally distributed population. Distribution of ages HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 13

14 Suppose we have a sample of n observations. Let's prepare a bar chart or a histogram of the sample depending on the type of the variable. In both cases, we have frequencies of categories or frequencies in the interval. Let's denote the frequency in the i-th category or interval by k i, i1,,,r (r is the number of categories). Let's denote p i the probabilities of falling into a given category or interval in the case of the given distribution. If H0 is true and n is large, then the relative frequencies are approximations of p i -s, or. ki n p i k np The formula of the test statistic has χ distribution with (r-1-s) degrees of freedom. Here s is the number Observed of the parameters frequency of the Expected distribution frequency (if there are). i i X r r ( Oi Ei ) ( ki n pi ) i 1 Ei i 1 n pi HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 14

15 Test for uniform distribution Example. We would like to test whether a dice is fair or biased. The dice is thrown 10 times. H0: the dice is fair, the probability of each category, p i 1/6. Calculation of expected frequencies: n p i 10 1/6 0. If it is fair, every throwing are equally probable so in ideal case we would expect 0 frequencies for each number Observed frequencies Expected frequencies X 6 ( i 1 k i 0) 0 1 [(5 0) + (18 0) + (1 0) 0 1 ( ) ) + (0 0) + (19 0) The degrees of freedom is 5, the critical value in the table is As our test statistic, < we do not reject H0 and claim that the dice is fair. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 15

16 Test for uniform distribution Example. We would like to test whether a dice is fair or biased. The dice is thrown 10 times. H0: the dice is fair, the probability of each category, p i 1/6. Calculation of expected frequencies: n p i 10 1/6 0. If it is fair, every throwing are equally probable so in ideal case we would expect 0 frequencies for each number Observed frequencies Expected frequencies X 6 ( i 1 k i 0) 0 1 [(5 0) + (18 0) + (1 0) 0 1 ( ) (17 0) + (0 0) + (39 0) The degrees of freedom is 5, the critical value in the table is As our test statistic, 30 > we reject H0 and claim that the dice is not fair. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 16

17 Test for normality Let's suppose we have a sample and would like to know whether it comes from a normally distributed population. H0: the sample is drawn from a normally distributed population. Let's make a histogram from the sample, so we get the "observed" frequencies. To test the null hypothesis we need the expected frequencies. We have to estimate the parameters of the normal density functions. We use the sample mean and sample standard deviation. The expected frequencies can be computed using the tables of the normal distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 17

18 30 Body height X r ki npi ( ) np i 1 i 0 10 k i Frequency Std. Dev 8.5 Mean N np i Body height HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 18

19 Using Gauss-paper There is a graphical method to check normality. The "Gauss-paper" is a special coordinate system, the tick marks of the y axis are the inverse of the normal distribution and are given in percentages. We simply have to draw the distribution function of the sample into this paper. In the case of normality the points are arranged approximately in a straight line. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 19

20 SPSS: Q-Q plot (quantile-quantile plot) HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 0

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