Topic 21: Goodness of Fit

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1 Topic 21: December 5, 2011 A goodness of fit tests examine the case of a sequence of independent observations each of which can have 1 of k possible categories. For example, each of us has one of 4 possible of blood types, A, B, AB, and O. The local blood bank has good information from a national database of the fraction of individuals having each blood type, π O, π A, π B, and π AB. The actual fraction p O, p A, p B, and p AB of these blood types in the community for a given blood bank may be different than what is seen in the national database. As a consequency, the local blood bank may choose to alter its distribution of blood supply to more accurately reflect local conditions. To place this assessment strategy in terms of formal hypothesis testing, let π = (π 1,..., π k ) be postulated values of the probability P π {individual is a member of i-th category} = π i and let p = (p 1,..., p k ) denote the possible states of nature. Then, the parameter space is The hypothesis test is Θ = {p = (p 1,..., p k ); p i 0 for all i = 1,..., k, p i = 1}. H 0 : p i = π i, for all i = 1,..., k versus H 1 : p i π i, for some i = 1,..., k. The data x = (x 1,..., x n ) are the outcomes of the n observations. Let s use the likelihood ration criterion to create a test for the distribution of human blood types in a given population. For the data x = {O, B, O, A, A, A, A, A, O, AB} for the blood types of tested individuals, then, in the case of independent observations, the likelihood is L(p x) = p O p B p O p A p A p A p A p A p O p AB = p 3 Op 5 Ap B p AB. Notice that the likelihood has a factor of p i whenever an observation take on the value i. In other words, if we summarize the data using n i = #{observations from category i} to create n = (n 1, n 2,, n k ), a vector that records the number of observations in each category, then, the likelihood function L(p n) = p n1 1 pn k k. The likelihood ratio is the ratio of the maximum value of the likelihood under the null hypothesis and the maximum likelihood for any parameter value. In this case, the numerator is the likelihood evaluated at π. Λ(n) = L(n π) L(n ˆp) = πn1 1 πn2 2 ( ) πn k n1 ( ) nk k π1 πk ˆp n1 1 ˆpn2 2 =. (1) ˆpn k ˆp 1 ˆp k k 261

2 To find the maximum likelihood estimator ˆp, we, as usual, begin by taking the logarithm ln L(p n) = n i ln p i. Because not every set of values for p i is admissible, we cannot just take derivatives, set them equal to 0 and solve. Indeed, we must find a maximum under the constraint s(p) = p i = 1. The maximization problem is now stated in terms of the method of Lagrange multipliers. This method tells us that at the maximum likelihood estimator (ˆp 1,..., ˆp k ), the gradient of ln L(p n) is proportional to the gradient of the constraint s(p). To explain this briefly, recall that the gradient of a function is a vector that is perpendicular to a level set of that function. In this case, ˆp s(p) is perpendicular to the level set {p; s(p) = 1}. Now imagine walking along the set of parameter values of p given by the constraint s(p) = 1, keeping track of the values of the score function ln L(p n). If the walk takes us from a value of the score function below l 0 to values above l 0 then (See Figure 1.), the level surfaces s(p) = 1 {p; s(p) = 1} and {ln L(p n = l 0 } intersect. Consequently, the gradients level sets for the score function, values increasing moving downward ˆp s(p) and ˆp ln L(p n) point in different directions on the intersection of these two surfaces. At Figure 1: Lagrance multipliers Level sets of the score function.shown in dashed blue. The level a local maximum or minimum of the set {s(p) = 1} shown in black. The gradients for the score function and the constraint indicated score function, the level surfaces are by dashed blue and black arrows, respectively. At the maximum, these two arrows are parallel, Their ratio λ is called the Langrange multiplier. If we view the blue dashed lines as contour lines tangent and the two gradients are parallel. In other words, the these two up or down hill. When the trail reaches its highest elevation, the trail is tangent to a contour line on a contour map and the black line as a trail. The crossing contour line indicates walking either gradients vectors are related by a constant of proportionality, λ, known as the Lagrange multiplier. Consequently, at extreme and the gradient for the hill is perpendicular to the trail. values, ( p 1 ln L(ˆp n),..., p ln L(ˆp n) = λ ˆp s(p). ) ln L(ˆp n) = p k ( n1,..., n ) k = λ(1,..., 1) ˆp 1 ˆp k ( p 1 s(p),..., Each of the components of the two vectors must be equal. In other words, ) s(p) p k n i ˆp i = λ, n i = λˆp i for all i = 1,..., k. (2) 262

3 Now sum this equality for all values of i and use the constraint s(p) = 1 to obtain Returning to (2), we have that n i = λ ˆp i = λs(p) = λ and n = λ. n 1 ˆp i = n and ˆp i = n i n. This is the answer we would guess - the estimate for p i is the fraction of observations in category i. We substitute for ˆp i in the likelihood ratio (1) to obtain Λ(n) = L(n π) L(n ˆp) = ( ) n1 ( ) nk nπ1 nπk. n 1 Recall that we reject the null hypothesis if this ratio is too low, i.e, the maximum likelihood under the null hypothesis is sufficiently smaller than the maximum likelihood under the alternative hypothesis. For random variables X 1, X 2,..., X n having distribution π, let s summarize the observations as suggested above N i = #{j; X j = i} Set N = (N 1,..., N k ) to be the vector of observed number of occurrences for each category i. Then the likelihood ratio test statistic 2 ln Λ n (N) = 2 N i ln nπ i = 2 N i ln N i N i nπ i has approximately a χ 2 k 1 distribution. The test statistic 2 ln Λ n (n) is generally rewritten using the notation O i = n i for the number of observed occurrences of i and E i = nπ i for the number of expected occurrences of i as given by H 0. Then, we can write the test statistic as 2 ln Λ n (O) = 2 O i ln O i (3) E i This is called the G test statistic. The traditional method for a test of goodness of fit was introduced between 1895 and 1900 by Karl Pearson and consequenttly has been in use for longer that the concept of likelihood ratio tests. To show the connection between the two tests, we begin by giving the quadratic Taylor polynomial approximation to the natural logarithm expanded around the value a 0 = 1. Apply this to 2 ln Λ n (n), we find that 2 ln Λ n (n) = 2 The data can be displayed in a table = 2 n k ln a (a 1) 1 2 (a 1)2. = 0 + ( (nπi ) n i 1 1 n i 2 (nπ i n i ) + (nπ i n i ) 2 n i. ( ) ) 2 nπi 1 n i ( ) 2 nπi n i 1 n i 263

4 Then, 2 ln Λ n (O) i 1 2 k observed O 1 O 2 O k expected E 1 E 2 E k (nπ i n i ) 2 n i (O i E i ) 2 E i. Example 1. The Red Cross recommends that a blood bank maintains 44% blood type O, 42% blood type A, 10% blood type B, 4% blood type AB. You suspect that the distribution of blood types in Tucson is not the same as the recommendation. In this case, the hypothesis is H 0 : p O = 0.44, p A = 0.42, p B = 0.10, p AB = 0.04 versus H 1 : at least one p i is unequal to the given values Based on 400 observations, we find 228 type O, 124 type A, 40 type B and 8 type AB. This give the table i O A B AB observed expected The commands to perform the chi-square test in R is shown below. The program computes the expected number of observations. > chisq.test(c(228,124,40,8),p=c(0.44,0.42,0.10,0.04)) Chi-squared test for given probabilities data: c(228, 124, 40, 8) X-squared = , df = 3, p-value = 8.977e-07 The number of degrees of freedom is 4 1 = 3. Note that the p-value is very low and so the distribution of blood types in Tucson is very unlikely to be the same as the national distribution. We can also perform the test using the G-statistic in (3): > O<-c(228,124,40,8) > E<-sum(O)*c(0.44,0.42,0.10,0.04) > Gstat<-2*sum(O*log(O/E)) > Gstat [1] > 1-pchisq(Gstat,3) [1] e-07 Example 2 (Hardy-Weinberg equilibrium). As we saw with Gregor Mendel s pea experiments, the two-allele Hardy- Weinberg principle states that after two generations of random mating the genotypic frequencies can be represented by a binomial distribution. So, if a population is segregating for two alleles A 1 and A 2 at an autosomal locus with frequencies p 1 and p 2. Then, random mating would give a proportion p 11 = p 2 1 for the A 1 A 1 genotype, p 12 = 2p 1 p 2 for the A 1 A 2 genotype, and p 22 = p 2 2 for the A 2 A 2 genotype. Then, p 1 = p p 12 p 2 = p p 12. This give us 5 parameters: p 1, p 2, p 11, p 12, p 22, with the constraints that p 1 + p 2 = p 11 + p 12 + p 22 = 1. Thus n = 3. The two restrictions from the Handy-Weinberg equilibrium above give us that k = 2. Thus, the chi-square test statistic will have 1 degree of freedom. 264

5 McDonald et al. (1996) examined variation at the CVJ5 locus in the American oyster, Crassostrea virginica. There were two alleles, L and S, and the genotype frequencies in Panacea, Florida were 14 LL, 21 LS, and 25 SS. So, So, the estimate of p 1 and p 2 are So, the expected number of observations is ˆp 11 = 14 60, ˆp 12 = 21 60, ˆp 22 = ˆp 1 = ˆp ˆp 12 = , ˆp 2 = ˆp ˆp 12 = E 11 = 60ˆp 2 1 = , E 12 = 60 2ˆp 1 ˆp 2 = , E 22 = 60ˆp 2 2 = The chi-square statistic The p-value (14 10) (21 29) (25 21)2 21 = = > 1-pchisq(4.569,1) [1] Thus, we have moderate evidence againt the null hypothesis of a Hardy-Weinberg equilibrium. Many forces may be the cause of this - non-random mating, selection, or migration to name a few possibilities. Exercise 3. Perform the chi square test using the G statistic for the example above. 1 Contingency tables For an r c contingency table, we consider two classifications A and B for an experiment. This gives c categories for classification A and r categories A 1,... A c B 1,... B r for classification B. Here, we write O ij to denote the number of occurences for which an individual falls into both category A i and category B j. The results is then organized into a two-way table. A 1 A 2 A c total B 1 O 11 O 12 O 1c O 1 B 2 O 21 O 22 O 2c O B r O r1 O r2 O rc O r total O 1 O 2 O c n The null hypothesis is that the classifications A and B are independent. To set the parameters for this model, we define p ij = P {an individual is simultaneously a member of category A i and category B j }. 265

6 Then, we have the parameter space Θ = {p = (p ij, 1 i r, 1 j c); p ij 0 for all i, j = 1, r c p ij = 1}. j=1 Write the marginal probabilities c r p i = p ij and p j = p ij. j=1 The null hypothesis of independence of the categories A and B can be written H 0 : p ij = p i p j, for all i, j versus H 1 : p ij p i p j, for some i, j. Follow the procedure as before for the goodness of fit test to end with the test statistic 2 r c j=1 O ij ln E ij O ij r c j=1 (O ij E ij ) 2 E ij. The null hypothesis p ij = p i p j can be written in terms of observed and expected observations as or E ij n = O i O j n n. E ij = O i O j n. Example 4. Returning to the study of the smoking habits of 5375 high school children in Tucson in 1967, here is a two-way table summarizing some of the results. The expected table is student student smokes does not smoke total 2 parents smoke parent smokes parents smoke total student student smokes does not smoke total 2 parents smoke parent smokes parents smoke total For example, E 11 = O 1 O 1 n = =

7 To compute the chi-square statistic ( ) ( ) ( ) ( ) ( ) ( ) = = The number of degrees of freedom is (r 1) (c 1) = (3 1) (2 1) = 2. Thus, the p-value > 1-pchisq(37.57,2) [1] e-09 is very small and leads us to reject the null hypothesis. Thus, children smoking habits are not independent of their parents smoking habits. An examination of the individual cells shows that the children of parents who do not smoke are less likely to smoke and children who have two parents that smoke are more likely to smoke. Under the null hypothesis, each cell has a mean of 1 and so values much greater than 1 show contribution that leads to the rejection of H 0. R does the computation for us using the chisq.test command > smoking<-matrix(c(400,416,188,1380,1823,1168),nrow=3) > smoking [,1] [,2] [1,] [2,] [3,] > chisq.test(smoking) Pearson s Chi-squared test data: smoking X-squared = , df = 2, p-value = 6.959e-09 Example 5. We have the following data on malaria in three regions of the world. The expected table is South Asia America Africa Total Malaria Type A Malaria Type B Malaria Type C Total

8 South Asia America Africa Total Malaria Type A Malaria Type B Malaria Type C Total The values for each of the cels in the chi-square statistic is = The degrees of freedom are (c 1)(r 1) = 2 2 = 4. In R, we have > malaria<-matrix(c(31,2,53,45,53,2,14,5,45),nrow=3,ncol=3) > malaria [,1] [,2] [,3] [1,] [2,] [3,] > chisq.test(malaria) Pearson s Chi-squared test data: malaria X-squared = , df = 4, p-value < 2.2e-16 This provides strong evidence against the independence of malaria type and continent. Exercise 6 (two-by-two tables). Here is the contingency table can be thought of as two sets of Bernoulli trials. group 1 group 2 total successes x 1 x 2 x 1 + x 2 failures n 1 x 1 n 2 x 2 (n 1 + n 2 ) (x 1 + x 2 ) total n 1 n 2 n 1 + n 2 Show that the chi-square test is equivalent to the two-sided two sample proportion test. 2 Applicability and Alternatives to Chi-squared Tests The chi-square test uses the central limit theorem and so is based on the ability to us a normal approximation. On criterion, the Cochran conditions requires no cell has count zero, and more than 80% of the cells have counts at least 5. If this does not hold, then Fisher s exact test uses the hypergeometric distribution (or its generalization) directly rather than normal approximation. For example, for the 2 2 table 268

9 Then, from the point of view of the B classification: A 1 A 2 total B 1 O 11 O 12 O 1 B 2 O 21 O 22 O 2 total O 1 O 2 n From the O 1 belonging to category B 1, O 11 also belong to A 1 This outcome can be accomplished in ) ( O1 ways. From the O 2 belonging to category B 2, O 12 also belong to A 1 This outcome can be accomplished in ) O 11 ( O2 O 21 We have O 1 from n observations with A 1. This can be accomplished in ( ) O 1 n ways. Under the null hypothesis that every individual can be placed in any group, provided we have the given marginal information. In this case, the probability of the table above has hypergeometric formula ( O1 )( O2 ) O 11 O ( 21 O 1 n ) = O 1!O 2!O 1!O 2! O 11!O 12!O 21!O 22!n!. (4) Notice that the formula is symmetric in the column and row variables. Thus, if we had derived the hypergeometric formula from the point of view of the A classification we would have obtained exactly the dame formula (4). To complete the exact test, we now compute the hypergeometric probabilities over all possible choices for entries in the cells that result in the given marginal values, and rank these probabilities from most likely to least likely. Find the ranking of the actual data. For a one-sided test of too rare, the P -value is the sum of probabilities of the ranking lower than that of the data. A similar idea applies to general r c tables. Example 7. We now return to a table on hemoglobin genotypes on two Indonesian islands. Recall that heterozygotes are protected against malaria. genotype AA AE EE Flores Sumba We noted that heterozygotes are rare on Flores and that it appears that malaria is less prevalent there since the heterozygote does not provide an adaptive advantage. Here are both the chisquare test and the Fisher exact test. 269

10 > genotype<-matrix(c(128,119,6,78,0,4),nrow=2) > genotype [,1] [,2] [,3] [1,] [2,] > chisq.test(genotype) Pearson s Chi-squared test data: genotype X-squared = , df = 2, p-value = 1.238e-12 Warning message: In chisq.test(genotype) : Chi-squared approximation may be incorrect > fisher.test(genotype) Fisher s Exact Test for Count Data data: genotype p-value = 3.907e-15 alternative hypothesis: two.sided Note that R cautions against the use of the chisquare test with these data. 3 Answer to Selected Exercise 3. Here is the R output. > O<-c(14,21,25) > phat<-c(o[1]+o[2]/2,o[3]+o[2]/2)/sum(o) > phat [1] > E<-sum(O)*c(phat[1]ˆ2,2*phat[1]*phat[2],phat[2]ˆ2) > E [1] > sum(e) [1] 60 > Gstat<-2*sum(O*log(O/E)) > Gstat [1] > 1-pchisq(Gstat,1) [1] The table of expected observations successes group 1 group 2 total n 1(x 1+x 2) n 2(x 1+x 2) n 1+n 2 n 1+n 2 x 1 + x 2 n failures 1((n 1+n 2) (x 1+x 2)) n 2((n 1+n 2) (x 1+x 2)) n 1+n 2 n 1+n 2 (n 1 + n 2 ) (x 1 + x 2 ) total n 1 n 2 n 1 + n 2 270

11 Now, write ˆp i = x i /n i for the sample proportions from each group, and ˆp 0 = x 1 + x 2 n 1 + n 2 for the pooled sample proportion. Then we have the table of observed and expected observations observed group 1 group 2 total successes n 1 ˆp 1 n 2 ˆp 2 (n 1 + n 2 )ˆp 0 failures n 1 (1 ˆp 1 ) n 2 (1 ˆp 2 ) (n 1 + n 2 )(1 ˆp 0 ) total n 1 n 2 n 1 + n 2 The chi-square test statistic expected group 1 group 2 total successes n 1 ˆp 0 n 2 ˆp 0 (n 1 + n 2 )ˆp 0 failures n 1 (1 ˆp 0 ) n 2 (1 ˆp 0 ) (n 1 + n 2 )(1 ˆp 0 ) total n 1 n 2 n 1 + n 2 (n 1(ˆp 1 ˆp 0)) 2 n 1 ˆp 0 + (n2(ˆp2 ˆp0))2 n 2 ˆp 0 + (n1((1 ˆp1) (1 ˆp0)))2 n 1(1 ˆp 0) + (n2((1 ˆp2)+(1 ˆp0)))2 n 2(1 ˆp 0) = n 1 (ˆp 1 ˆp 0) 2 ˆp 0 + n 2 (ˆp 2 ˆp 0) 2 ˆp 0 (ˆp + n 1 ˆp 0) 2 (ˆp 1 (1 ˆp 0) + n 2 ˆp 0) 2 2 (1 ˆp 0) = n 1 (ˆp 1 ˆp 0 ) 2 1 ˆp + n 0(1 p 0) 2(ˆp 2 ˆp 0 ) 2 1 n = 1(ˆp 1 ˆp 0) 2 +n 2(ˆp 2 ˆp 0) 2 ˆp 0(1 p 0) = ln Λ(x 1, x 2 ) from the likelihood ratio computation for the two-sided two sample proportion test ˆp 0(1 p 0) 271

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