Chi2 Tests

1
Chi2 Tests

• Recall that the Chi2 distn with dfN can be
  defined as the sum of squares of N ind. standard
  normals (p. 252)
• The Chi2 distn is a special form of the Gamma
  distn (p. 198)
• Write as ?2N
• Mean(?2N) N
• SD(?2N) ?(2N)

2
Chi2 Tests

• To compare two things (means or proportions), we
  can subtract
• To compare more than 2 things, we cant use
  subtraction
• Can use the sum of squares
• Sum of squares is used in the sample SD
  (variance) to measure how spread out the data
  is
• We can use SS to measure how spread out several
  probabilities are, for instance

3
Chi2 Tests

• A number of ?2 tests use the following formula
• ? (obs-exp)2/exp
• For a number of different reasons, this has
  (approximately) a ?2N distn
• Often a result of the Central Limit Thm causing
  obs to be (approx) normal
• The exact defn of obs and exp varies from problem
  to problem

4
Chi2 Tests

• In SM239, we developed the distn for the
  difference of two proportions (normal)
• Suppose we need to extend this to determining
  whether 3 or more probabilities are the same
• Need ?2

5
Chi2 Tests

• See p. 472, Ex29 and Fig 10.28
• Compare level of allergic reaction for 3 drugs

6
Chi2 Tests
7
Chi2 Tests

• Matlab note
• The Excel files have the data in one long column
• Call it x
• x2zeros(4,3) x2()x
• This will make it into a matrix
• BE SURE TO USE THE RIGHT SHAPE
• If we had used x2zeros(3,4), it would have
  produced a matrix, but the wrong one.
• The first argument is how often each category is
  repeated
• The second argument is how many catetories
  (Drugs) there are

8
Chi2 Tests

• H0 Reaction levels are the same for each drug
• Since we had equal numbers of the 3 drugs, we
  would expect equal numbers in each level of
  reaction
• Does not say that each level occurs ¼ of the time
• Expected should be 1/3 of the total in each
  category, since 1/3 of the people were given each
  drug

9
Chi2 Tests

• gtgt obs11 30 36 23
• 8 31 25 36
• 13 28 28 31
• gtgt csumsum(obs)
• gtgt rsum(sum(obs'))
• gtgt exptrsumcsum/sum(rsum)

10
Chi2 Tests

• Sum(rsum) is the total , 300 in this case
• Rsum/N is the fraction in each row
• Mult by Csum to get expected in each cell
• Rsum, Csum are the marginals (often written in
  the margins)

11
Chi2 Tests

• Then compute ?2
• gtgt c2a(obs-expt).2./expt
• c2a
• 0.0104 0.0037 1.3521 1.6333
• 0.6667 0.0599 0.7341 1.2000
• 0.5104 0.0936 0.0936 0.0333
• gtgt sum(sum(c2a))
• 6.3912

12
Chi2 Tests

• Exercises compute Chi2
• P. 472, Fig 10.29-30

13
Chi2 Tests

• Need df
• For these problems, df( rows-1)( cols-1)
• Df(3-1)(4-1) 6
• Can think of it as the number of relevant numbers
  that can vary in the problem
• Margins are not relevant
• Also, one row and one col cannot vary (determined
  by margins)
• Or just learn the formula

14
Chi2 Tests

• To compute p-value, have to determine more
  extreme
• If obsexpt, then it appears that H0 is true
• This would be when ?2 is small
• More extreme would be larger values
• I.e., prob (6.3912 or greater)

15
Chi2 Tests

• function ychiprob(df,lo,hi),
• gtgt chiprob(6,6.3912,99)
• 0.3808
• This is not a small probability
• Would not reject H0
• Data does not dispute the reaction levels being
  the same for all 3 drugs

16
Chi2 Tests

• For significance testing, how large would
  statistic have to be?
• For ?0.05,
• gtgt bisect(_at_(x) chiprob(6,x,99),0.05,6,99,.0001)
• 12.5917

17
Chi2 Tests

• Exercises
• P. 472, Fig 10.29-30
• Also p. 480, 10.4.7, 10.4.9, 10.4.10

18
Chi2 Tests

• Consider 10.2.3, p 455
• System A 35/44 successful
• B 36/52
• Test H0 p1p2 vs Ha not equal
• gtgt f135/44f236/52f0(3536)/(4452)
• gtgt pv2nprob(0,sqrt(f0(1-f0)(1/441/52)),f1-f2,
  99)
• 0.2512

19
Chi2 Tests

• Or, make a table

20
Chi2 Tests

• d
• 35 36
• 9 16
• gtgt yrbyc(d)
• 1.3166
• gtgt chiprob(1,y,99)
• 0.2512
• gtgt
• So we get the same (two-sided) answer by either
  method

21
Chi2 Tests

• Recall that Chi2 arises as sums of squares of
  normals
• CLT says that many quantities are approximately
  normal
• Guideline Need all the expected values to be at
  least 5 for normal approx and, hence, for Chi2

22
Chi2 Tests

• Consider 10.4.5, p 480 (DS10.4.5)
• Confirm that ALL expected values for Not
  Satisfied are lt5
• According to guideline, Chi2 is not appropriate
  here

23
Chi2 Tests

• Small sample problem can be solved using Fishers
  Exact Test
• Based on hypergeometric distn
• Wont deal with here

24
Chi2 Tests

• Could express hypotheses in terms of independence
• H0 rows and cols are independent
• Ha probabilities not all equal
• Ha might have to say probability distributions
  not all equal

25
Chi2 Tests

• Not all the same does not mean all are
  different
• Could have some the same and some different

26
Chi2 Tests

• A new use for hypothesis tests
• Tells us something about the underlying process
• Suppose grade distn classes meeting at different
  times are the same
• I.e., grades are ind of time
• Tells us about the relationship (or lack) between
  grades and time

27
Chi2 Tests

• What about when we reject H0?
• Might like to know more about where the
  differences lie
• Not always a simple answer
• Sometimes, the differences can be clear

28
Chi2 Tests

• Suppose the drug data had been

29
Chi2 Tests
30
Chi2 Tests

• Chi2 value is 36.12, which is very large
• The terms in Chi2 are

31
Chi2 Tests
32
Chi2 Tests

• Suggest that No for Drug C is quite different
  from what we expected
• Can pool to test this
• BIG SECRET The df are not changed when we pool
  as a result of how the data looked

33
Chi2 Tests

• Pooling programs
• Cpool(table,c1,c2)
• Rpool(table,r1,r2)
• Adds 2 cols/rows and zeros the second one
• Have to write chi2() so you dont get divide by
  zero
• c2a(obs-expt).2./max(expt,expt0)

34
Chi2 Tests

• gtgt o2rpool(obs,1,2)
• 19 70 71 40
• 0 0 0 0
• 13 19 18 50
• gtgt y,expt,c2a,rsum,csumrbyc(o2),
• y
• 33.4005

35
Chi2 Tests

• Compare Chi2 for pooled table (33.4005) to
  original value (36.1246)
• Can show that Chi2 cannot increase (and almost
  always goes down) after pooling
• Trick is that it doesnt go down very much if you
  do it right
• Remember that df are not changing, so if Chi2
  goes down too much, it will no longer be
  significant

36
Chi2 Tests

• gtgt o3cpool(o2,2,1)
• 0 89 71 40
• 0 0 0 0
• 0 32 18 50
• gtgt yrbyc(o3)
• 29.4647
• Still quite large

37
Chi2 Tests

• gtgt o4cpool(o3,3,2)
• 0 0 160 40
• 0 0 0 0
• 0 0 50 50
• gtgt yrbyc(o4)
• 28.5714
• Still very large
• Cannot pool any more
• Conclude that the differences in drug reactions
  are due to No reaction to Drug C

38
Chi2 Tests

• It is possible to come to different conclusions
  from the same data
• Depends on what you choose to pool
• Can be guided by what makes sense
• Also look at (obs-expt)./expt (no 2)
• This will tell which cells are above and below
  expt and about how much