Topic 8. Chi Square Tests

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1 BE540W Chi Square Tests Page 1 of 5 Topic 8 Chi Square Tests Topics 1. Introduction to Contingency Tables. Introduction to the Contingency Table Hypothesis Test of No Association.. 3. The Chi Square Test of No Association in an R x C Table (For Epidemiologists) Special Case: More on the x Table. 5. Hypotheses of Independence or No Association. Appendix Relationship Between the Normal(0,1) and the Chi Square Distribution

2 BE540W Chi Square Tests Page of 5 1. Introduction to Contingency Tables We wish to Explore the Association Between Two Variables, Both Discrete Example - Is smoking associated with low birth weight? A common goal of many research studies is investigation of the association of factors, both discrete; eg smoking (yes/no) and low birth weight (yes/no). In Topic 8, our focus is in the setting of two categorical variables, such as smoking and low birth weight, and the use of chi-square tests of association and homogeneity. Introduction to Contingency Tables Example - Suppose we do a study to investigate the relationship between smoking and impairment of lung function, measured by forced vital capacity (FVC). Suppose n = 100 people are selected for the study. For each person we note their smoking behavior (smoke or don t smoke) and their forced vital capacity, FVC (normal or abnormal). FVC

3 BE540W Chi Square Tests Page 3 of 5 ne scenario is the following set of counts All 50 smokers have an abnormal FVC And all 50 non-smokers have normal FVC This is an illustration of a perfect association in that once smoking status is known, FVC status is known also Another scenario is the following set of counts In this scenario, half (5) of the smokers have an abnormal FVC But we also observe that half (5) of the 50 non-smokers have an abnormal FVC, also. This similarity in the data suggest that there is no association between smoking status and FVC Put another way, the data suggest that lung function, as measured by FVC, is independent of smoking status.

4 BE540W Chi Square Tests Page 4 of 5. Introduction to the Contingency Table Hypothesis Test of No Association In Topic 7 (Hypothesis Testing), we used the idea of proof by contradiction to develop hypothesis tests. We do this in the contingency table setting, too. Example, continued - Suppose we do a study to investigate the relationship between smoking and impairment of lung function, measured by forced vital capacity (FVC). What are our null and alternative hypotheses? Consider the notation that says π1 = the proportion of smokers with abnormal fvc π = the proportion of non-smokers with abnormal fvc The null hypothesis is that of independence, N association, and says the proportion with abnormal fvc is the same, regardless of smoking status. π1 = π The alternative hypothesis is that of association/dependence and says the proportion with abnormal fvc will be a different number, depending on smoking status. Thus, π1 π Ho: There is no association between the two variables, π1 = π we ll argue proof-by-contradiction from here Ha: The two variables are associated π1 π so as to advance this hypothesis

5 BE540W Chi Square Tests Page 5 of 5 Recall also from Topic 7 that a test statistic (also called pivotal quantity) is a comparison of what the data are to what we expected under the assumption that the null hypothesis is correct Introduction to observed versus expected counts. bserved counts are represented using the notation or n. Expected counts are represented using the notation E FVC Abnormal Normal Smoke Don t smoke How to read the subscripts - o The first subscript tells you the row ( e.g. 1 is a cell count in row ) o The second subscript tells you column ( e.g. 1 is a cell count in col 1 ) o Thus, 1 is the count for the cell in row and column 1 o A subscript replaced with a dot is tells you what has been totaled over. Thus,. is the total for row taken over all the columns Similarly.1 is the total for column 1, taken over all the rows And.. is the grand total taken over all rows and over all columns

6 BE540W Chi Square Tests Page 6 of 5 Here are the observed counts in another scenario Smoke Don t smoke FVC Abnormal Normal 11=40 1 =10 1. =50 1=5 =45. =50.1=45. =55.. =100 = 5 is # in row column 1 1 = 10 is # in row 1 column = 50 is the row 1 total.1= 45 is the column 1 total 1 1. What are the expected counts E under the assumption that the null hypothesis is true? Hint You already have an intuition for this. If a fair coin is tossed 0 times, the expected number of heads is represents (0 tosses) x (.50 probability of heads on each toss) = 10. Solution: Recall that we are utilizing the notation that says π1 = the proportion of smokers with abnormal fvc π = the proportion of non-smokers with abnormal fvc Under the assumption that the null of N association/independence, then π1 = π = π a common (null hypothesis) value The common π is estimated as the observed overall proportion of abnormal fvc. Hint You already know this intuitively as = 45/100 = 0.45 = (total # with abnormal fvc)/(grand total). 45 column 1 total ˆπ = = 100 grand total, or a bit more formally ˆ= = = = π

7 BE540W Chi Square Tests Page 7 of 5 Thus, under the assumption that Ho is true (meaning no association, independence), the proportion with abnormal fvc among smokers as well as among non-smokers should be the same as in the overall population, that is, π 1;null = π ;null = ˆπ = 0.45 So we expect 45% of the 50 smokers, or.5 persons, to have abnormal fvc, and we also expect 45% of the non-smokers, or.5 persons, to have abnormal fvc. Yes, you are right. These expected counts are NT whole integers. That s okay. Do NT round. Expected # smokers w abnormal FVC = (#Smokers)(π) ˆ = (50)(.45) =.5 = E 11 Expected # NNsmokers w abnormal FVC = (#NNSmokers)(π) ˆ = (50)(.45) =.5 = E 1 We also need to obtain the expected counts of normal fvc. o We expect 55% of the 50 smokers, or 7.5, to have normal fvc, and we also expect 55% of the non-smokers, or 7.5, to have normal fvc. Expected # smokers w no rmal FVC = (#Smokers)(1-π) ˆ = (50)(.55) = 7.5 = E1 Expected # NNsmokers w normal FVC = (#NNSmokers)(1-π) ˆ = (50)(.55) = 7.5 = E Thus the following expected counts emerge. Smoke Don t smoke FVC Abnormal Normal E 11=.5 E 1 =7.5 E 1. =50 E 1=.5 E =7.5 E. =50 E.1=45 E. =55 E.. =100 E =.5 E 1 =7.5 E =50 E.1=

8 BE540W Chi Square Tests Page 8 of 5 IMPRTANT NTE about row totals and column totals - o The expected row totals match the observed row totals. o The expected column totals match the observed column totals. o These totals have a special name - marginals. The marginals are treated as fixed constants ( givens ).

9 BE540W Chi Square Tests Page 9 of 5 What is the test statistic (pivotal quantity)? It is a chi square statistic. Examination reveals that it is defined to be a function of the comparisons of observed and expected counts. It can also be appreciated as a kind of signal-tonoise construction. Chi Square df = = χ df all cells "i,j" ( ) E E = all cells "i,j" ( bserved ) Expected Expected When the null hypothesis of no association is true, the observed and expected counts will be similar, their difference will be close to zero, resulting in a SMALL chi square statistic value. Example A null hypothesis says that a coin is fair. Since a fair coin tossed 0 times is expected to yield 10 heads, the result of 0 tosses is likely to be a number of observed heads that is close to 10. When the alternative hypothesis of an association is true the observed counts will be unlike the expected counts, their difference will be non zero and their squared difference will be positive, resulting in a LARGE PSITIVE chi square statistic value. Example The alternative hypothesis says that a coin is NT fair. An UNfair coin tossed 0 times is expected to yield a different number of heads that is NT close to the null expectation value of 10. Thus, evidence for the rejection of the null hypothesis is reflected in LARGE PSITIVE values of the chi square statistic.

10 BE540W Chi Square Tests Page 10 of 5 3. The Chi Square Test of No Association in an R x C Table For reasons not detailed here (see Appendix), the comparison of observed and expected counts defined on page 9 is, often, distributed chi square when the null is true. For one cell, when the null is true, L NM bserved Expected Count - Count Expected Count QP is distributed Chi Square (df = 1) approximately. Summed over all cells in an R x C table, when the null is true, In a table that has R rows and C columns, the same calculation is repeated RC times and then summed R i= 1 C j= 1 L NM bserved Expected Count (i, j) - Count (i, j) Expected Count (i, j) QP is distributed Chi Square (df = [R-1][C-1]) approximately.

11 BE540W Chi Square Tests Page 11 of 5 More on Degrees of Freedom (What is the correct degrees of freedom?) Recall that degrees of freedom can be appreciated as the number of independent pieces of information. Here is how the idea works in contingency tables: In a x table Now see what happens in larger tables In each scenario, the last column is not free and the last row is not free.

12 BE540W Chi Square Tests Page 1 of 5 More generally, In an R x C table Degrees of freedom = (#rows 1) x (#columns 1) = (R 1)(C 1) We have the tools for computing the chi square test of association in a contingency table. Example Suppose we wish to investigate whether or not there is an association between income level and how regularly a person visits his or her doctor. Consider the following count data. Last Consulted Physician Income < 6 months 7-1 months >1 months Total < \$ \$6000-\$ \$10,000-\$13, \$14,000-\$19, > \$0, Total The general layout uses either the n or notation to represent the observed counts: Columns, j j = 1 j = C Rows, I i = 1 11 =n 11 1C =n 1C N 1. = 1. i = R R1 =n R1 RC =n RC N R. = R. N.1 =.1 N.C =.C N=..

13 BE540W Chi Square Tests Page 13 of 5 What are the π now? π = the probability of being the combination that is income at level i and time since last consult at level j Example: π 11 = probability [ income is <\$6000 AND time since last visit is < 6 mos] π i. = the overall (marginal) probability that income is at level i Example: π 1. = probability [ income is <\$6000 ] π.j = the overall (marginal) probability that time since last visit is at level j Example: π.1 = probability [ time since last visit is < 6 months ] What is independence now? Again, you already have an intuition for this. Recall the example of tossing a fair coin two times. Because the outcomes of the two tosses are independent, Probability of heads on toss 1 and heads on toss = (.50)(.50) =.5 Now attach some notation to this intuition. π 1. = Probability of heads on toss 1 π. = Probability of heads on toss π 1 = Probability of heads on toss 1 and heads on toss Independence π 1 = [ probability heads on toss 1 ] x [ probability heads on toss ] = [ π 1. ] [ π. ] Thus, under independence π = [ π i. ] [ π.j ] Pr[ i x j combination ] = [Marginal i prob] x [Marginal j ]

14 BE540W Chi Square Tests Page 14 of 5 Under independence π = [ π i. ] [π.j ] Example, continued- π i. = Probability that income is level i π.j = Probability that time since last visit is at level j π = Probability income is level i AND time since last visit is at level j Under Independence, π = [ π i. ] [ π.j ] Assumptions of Chi Square Test of N Association 1. The contingency table of count data is a random sample from some population. The cross-classification of each individual is independent of the cross-classification of all other individuals. Null and Alternative Hypotheses H : π = ππ i..j H : π ππ A i..j Null Hypothesis Estimates of the π ˆ π ˆ ππˆ = by independence and where i..j ˆ n n i. π i. = = row "i" total grand total ˆ n.j π.j = = n column "j" total grand total

15 BE540W Chi Square Tests Page 15 of 5 Null Hypothesis Expected Counts E [row "i" total][column "j" total] E = (# trials)[ ˆ π ˆ ˆ under null]=(n) πi. π.j = n Test Statistic (Pivotal Quantity) Just as we did before For each cell, departure of the observed counts from the null hypothesis expected counts is obtained using: c E E h The chi square test statistic of association is the sum of these over all the cells in the table: Test Statistic = R C ( ) E E i= 1 j= 1 Behavior of the Test Statistic under the assumption of the null hypothesis Under the null hypothesis Test Statistic = ( -E) R C is distributed χ df = (R-1)(C-1) i=1 j=1 E

16 BE540W Chi Square Tests Page 16 of 5 Decision Rule The null hypothesis is rejected for large values of the test statistic. Thus, evidence for rejection of the null hypothesis is reflected in the following (all will occur) LARGE value of test statistic SMALL value of achieved significance (p-value) - Test statistic value that EXCEEDS CRITICAL VALUE threshold Computations (1) For each cell, compute E [row "i" total][column "j" total] = n () For each cell, compute c E E h

17 BE540W Chi Square Tests Page 17 of 5 Example, continued - bserved Counts (this is just the table on page 1 again with the notation provided) Last Consulted Physician Income < 6 months 7-1 months >1 months Total < \$ = =38 13 =35 1. =59 \$6000-\$ =7 =54 3 =45. =36 \$10,000-\$13, =19 3 =78 33 =78 3. =375 \$14,000-\$19, =355 4 =11 43 = =607 > \$0, =653 5 =85 53 =59 5. =1197 Total.1 =1640. =567.3 =557.. =764 Expected Counts note that each entry is (row total)(column total)/(grand total) Last Consulted Physician Income < 6 months 7-1 months >1 months Total < \$6000 ( 59)( 1640) E 11 = = E 1 =53.13 E 13 =5.19 E 1. = \$6000-\$9999 E 1 = E =66.87 E 3 =65.70 E. =36 \$10,000-\$13,999 E 31 =.50 E 3 =76.93 E 33 =75.57 E 3. =375 \$14,000-\$19,999 E 41 = E 4 =14.5 E 43 =1.3 E 4. =607 > \$0,000 E 51 =710.3 E 5 =45.55 ( 1197)( 557) E 53 = = 41. E 5. =1197 Total E.1 =1640 E. =567 E.3 =557 E.. =

18 BE540W Chi Square Tests Page 18 of 5 ( E ) ( ) ( ) χ( R 1)( C 1) = = = all cells E with degrees of freedom = (R-1)(C-1) = (5-1)(3-1) = 8 Statistical Decision Using Significance Level (p-value) Approach p-value = Probability [ Chi square with df=8 > ] < <.0001 Such an extremely small p-value is statistically significant Reject the null hypothesis. To use: enter values in the boxes for df and chi square value. Click on RIGHT arrow You should then see 0 in the box labeled probability ; thus, probability <<<.0001

19 BE540W Chi Square Tests Page 19 of 5 Statistical Decision Using Critical Region Approach with type I error=.05 The critical region approach with type I error =.05 tells us to reject the null hypothesis for values of the test statistic that are greater than the 95 th percentile. Equivalently, it tells us to reject the null hypothesis for values of the test statistic that are in the upper 5% of the distribution. χ = is our critical value..95;df=8 bserved statistic = >> Reject the null hypothesis. χ.95;df=8 = To use: Click the radio dial button for area to the left. Then enter values in the boxes for df and probability. Then click on LEFT arrow You should then see the value in to box labeled chi square value Example, continued - These data provide statistically significant evidence that time since last visit to the doctor is NT independent of income, that there is an association between income and frequency of visit to the doctor. Important note! What we ve learned is that there is an association, but not its nature. This will be considered further in PubHlth 640, Intermediate Biostatistics.

20 BE540W Chi Square Tests Page 0 of 5 4. (For Epidemiologists) Special Case: More on the x Table Many epidemiology texts use a different notation for representing counts in the same chi square test of no association in a x table. Counts are a, b, c, and d as follows. nd Classification Variable 1 1 st Classification 1 a b a + b c d c + d a + c b + d n The calculation for the chi square test that you ve learned as being given by χ = all cells ( ) E E is the same calculation as the following shortcut formula ( ) ( a+c)( b+d)( c+d)( a+b) n ad-bc χ = when the notation for the cell entries is the a, b, c, and d, above.

21 BE540W Chi Square Tests Page 1 of 5 5. Hypotheses of Independence or No Association Independence, No Association, Homogeneity of Proportions are alternative wordings for the same thing. For example, (1) Length of time since last visit to physician is independent of income means that income has no bearing on the elapsed time between visits to a physician. The expected elapsed time is the same regardless of income level. () There is no association between coffee consumption and lung cancer means that an individual s likelihood of lung cancer is not affected by his or her coffee consumption. (3) The equality of probability of success on treatment (experimental versus standard of care) in a randomized trial of two groups is a test of homogeneity of proportions. The hypotheses of independence, no association, homogeneity of proportions are equivalent wordings of the same null hypothesis in an analysis of contingency table data.

22 BE540W Chi Square Tests Page of 5 Appendix Relationship Between the Normal(0,1) and the Chi Square Distributions For the interested reader.. This appendix explains how it is reasonable to use a continuous probability model distribution (the chi square) for the analysis of discrete (counts) data, in particular, investigations of association in a contingency table. Previously (see Topic 6, Estimation), we obtained a chi square random variable when working with a function of the sample variance S. It is also possible to obtain a chi square random variable as the square of a Normal(0,1) variable. Recall that this is what we have so far IF THEN Has a Chi Square Distribution with DF = Z has a distribution that is Normal (0,1) Z 1 X has a distribution that is Normal (µ, σ ), so that { Z-score } 1 Z-score = X-µ σ X 1, X,, X n are each distributed Normal (µ, σ ) and are independent, so that X is Normal (µ, σ /n) and { Z-score } 1 Z-score= X- µ σ n X 1, X,, X n are each distributed Normal (µ, σ ) and are independent and we calculate S = n b i= 1 X-X n 1 g (n-1)s σ (n-1)

23 BE540W Chi Square Tests Page 3 of 5 ur new formulation of a chi square random variable comes from working with a Bernoulli, the sum of independent Bernoulli random variables, and the central limit theorem. What we get is a great result. The chi square distribution for a continuous random variable can be used as a good model for the analysis of discrete data, namely data in the form of counts. Z 1, Z,, Z n are each Bernoulli with probability of event = π. E[Z i] = µ = π Var[Z ] = σ = π(1 π) 1. The net number of events X= Z i i n i=1 is Binomial (N,π). We learned previously that the distribution of the average of the Z i is well described as Normal(µ, σ /n). Apply this notion here: By convention, n Zi X i= 1 Z= = = X n n 3. So perhaps the distribution of the sum is also well described as Normal. At least approximately If X is described well as Normal (µ, σ /n) Then X=nX is described well as Normal (nµ, nσ ) Exactly: X is distributed Binomial(n,π) Approximately: X is distributed Normal (nµ, nσ ) Where: µ = π and σ = π(1- π)

24 BE540W Chi Square Tests Page 4 of 5 Putting it all together IF THEN Comment X has a distribution that is Binomial (n,π) exactly X has a distribution that is Normal (nµ, nσ ) approximately, where µ = π σ = π(1-π) Z-score= X-E(X) SD(X) = X-n µ nσ = X-nπ n π (1- π ) is approx. Normal(0,1) { Z-score } has distribution that is well described as Chi Square with df = 1. We arrive at a continuous distribution model (chi square) approximation for count data.

25 BE540W Chi Square Tests Page 5 of 5 Thus, the { Z-score } that is distributed approximately Chi Square (df=1) is the (-E) /E introduced previously. Preliminaries X = bserved = nπ = "Expected" = E As n gets larger and larger nπ(1- π ) nπ( 1 ) = "Expected" = E Upon substitution, { } ( ) X-nπ X-nπ -E -E Z-Score = = = nπ(1-π) nπ(1) E E Thus, For one cell, when the null hypothesis is true, the central limit theorem gives us L NM bserved Expected Count - Count Expected Count QP is Chi Square (df = 1) approximately. For RC cells, when the null hypothesis is true, the central limit theorem and the definition of the chi square distribution give us bserved Count Expected - Count R C is Chi Square [df =(R-1)(C- 1)] approx. i= 1 J= 1 Expected Count

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