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```14
Goodness-of-Fit Tests
and Categorical Data
Analysis
14.1
Goodness-of-Fit Tests When
Category Probabilities
Are Completely Specified
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
A binomial experiment consists of a sequence of
independent trials in which each trial can result in one of
two possible outcomes: S (for success) and F (for failure).
The probability of success, denoted by p, is assumed to be
constant from trial to trial, and the number n of trials is fixed
at the outset of the experiment.
3
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
We already presented a large-sample z test for testing
H0: p = p0.
Notice that this null hypothesis specifies both P(S) and
P(F), since if
P(S) = p0, then P(F) = 1 – p0.
Denoting P(F) by q and 1 – p0 by q0, the null hypothesis
can alternatively be written as
H0: p = p0, q = q0.
4
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
The z test is two-tailed when the alternative of interest is
p ≠ p0. A multinomial experiment generalizes a binomial
experiment by allowing each trial to result in one of k
possible outcomes, where k > 2.
For example, suppose a store accepts three different types
of credit cards. A multinomial experiment would result from
observing the type of credit card used—type 1, type 2, or
type 3—by each of the next n customers who pay with a
credit card.
In general, we will refer to the k possible outcomes on any
given trial as categories, and pi will denote the probability
that a trial results in category i.
5
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
If the experiment consists of selecting n individuals or
objects from a population and categorizing each one, then
pi is the proportion of the population falling in the ith
category (such an experiment will be approximately
multinomial provided that n is much smaller than the
population size).
The null hypothesis of interest will specify the value of each
pi.
For example, in the case k = 3, we might have
H0: p1 = .5, p2 = .3, p3 = .2.
6
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
The alternative hypothesis will state that H0 is not
true—that is, that at least one of the pi’s has a value
different from that asserted by H0 (in which case at least
two must be different, since they sum to 1).
The symbol pi0 will represent the value of pi claimed by the
null hypothesis. In the example just given, p10 = .5, p20 = .3,
and p30 = .2.
Before the multinomial experiment is performed, the
number of trials that will result in category i(i = 1,2,…, or k)
is a random variable—just as the number of successes and
the number of failures in a binomial experiment are random
variables.
7
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
This random variable will be denoted by Ni and its
observed value by ni.
Since each trial results in exactly one of the k categories,
Ni = n, and the same is true of the ni’s. As an example, an
experiment with n = 100 and k = 3 might yield N1 = 46,
N2 = 35, and N3 = 19.
The expected number of successes and expected number
of failures in a binomial experiment are np and nq,
respectively.
8
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
When H0: p = p0, q = q0 is true, the expected numbers of
successes and failures are np0 and nq0, respectively.
Similarly, in a multinomial experiment the expected number
of trials resulting in category i is
E(Ni) = npi(i = 1,…, k).
When H0: p1 = p10,..., pk = pk0 is true, these expected values
become
E(N1) = np10, E(N2) = np20,…, E(Nk) = npk0.
9
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
For the case k = 3,
H0: p1 = .5,
p2 = .3, p3 = .2, and
n = 100, the expected frequencies when H0 is true are
E(N1) = 100(.5) = 50,
E(N2) = 30, and
E(N3) = 20.
10
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
The ni’s and corresponding expected frequencies are often
displayed in a tabular format as shown in Table 14.1.
Observed and Expected Cell Counts
Table 14.1
The expected values when H0 is true are displayed just
below the observed values.
11
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
The Ni’s and ni’s are usually referred to as observed cell
counts (or observed cell frequencies), and np10, np20,…,
npk0 are the corresponding expected cell counts under H0.
The ni’s should all be reasonably close to the
corresponding npi0’s when H0 is true.
On the other hand, several of the observed counts should
differ substantially from these expected counts when the
actual values of the pi’s differ markedly from what the null
hypothesis asserts.
12
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
The test procedure involves assessing the discrepancy
between the ni’s and the npi0’s, with H0 being rejected when
the discrepancy is sufficiently large.
It is natural to base a measure of discrepancy on the
squared deviations (n1 – np10)2, (n2 – np20)2,…, (nk – npk0)2.
A seemingly sensible way to combine these into an overall
measure is to add them together to obtain (ni – npi0)2.
However, suppose np10 = 100 and np20 = 10. Then if
n1 = 95 and n2 = 5, the two categories contribute the same
squared deviations to the proposed measure.
13
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
Yet n1 is only 5% less than what would be expected when
H0 is true, whereas n2 is 50% less.
To take relative magnitudes of the deviations into account,
each squared deviation is divided by the corresponding
expected count.
Before giving a more detailed description, we must discuss
a type of probability distribution called the chi-squared
distribution.
14
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
The chi-squared distribution has a single parameter n,
called the number of degrees of freedom (df) of the
distribution, with possible values 1, 2, 3,….
Analogous to the critical value t,v for the t distribution,
is the value such that  of the area under the X2 curve with
df lies to the right of
(see Figure 14.1).
A critical value for a chi-squared distribution
Figure 14.1
15
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
Selected values of
are given in Appendix Table A.7.
Theorem
Provided that npi  5 for every i (i = 1, 2,…, k), the variable
has approximately a chi-squared distribution with k – 1 df.
The fact that df = k – 1 is a consequence of the restriction
Ni = n.
16
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
Although there are k observed cell counts, once any k – 1
are known, the remaining one is uniquely determined.
That is, there are only k – 1 “freely determined” cell counts,
and thus k – 1 df.
If npi0 is substituted for npi in X2, the resulting test statistic
has a chi-squared distribution when H0 is true.
Rejection of H0 is appropriate when X2  c (because large
discrepancies between observed and expected counts lead
to a large value of X2), and the choice
yields a
test with significance level .
17
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
Null hypothesis: H0: p1 = p10, p2 = p20,…, pk = pk0
Alternative hypothesis: Ha: at least one pi does not equal pi0
Test statistic value:
Rejection region:
18
Example 1
If we focus on two different characteristics of an organism,
each controlled by a single gene, and cross a pure strain
having genotype AABB with a pure strain having genotype
aabb (capital letters denoting dominant alleles and small
letters recessive alleles), the resulting genotype will be
AaBb.
If these first-generation organisms are then crossed among
themselves (a dihybrid cross), there will be four phenotypes
depending on whether a dominant allele of either type is
present.
19
Example 1
cont’d
Mendel’s laws of inheritance imply that these four
phenotypes should have probabilities
and
arising in any given dihybrid cross.
of
The article “Linkage Studies of the Tomato” (Trans. Royal
Canadian Institute, 1931: 1–19) reports the following data
on phenotypes from a dihybrid cross of tall cut-leaf
tomatoes with dwarf potato-leaf tomatoes.
There are k = 4 categories corresponding to the four
possible phenotypes, with the null hypothesis being
H0: p1 =
, p2 =
, p3 =
, p4 =
20
Example 1
cont’d
The expected cell counts are 9n/16, 3n/16, 3n/16, and
n/16, and the test is based on k – 1 = 3 df.
The total sample size was n = 1611. Observed and
expected counts are given in Table 14.2.
Observed and Expected Cell Counts for Example 1
Table 14.2
21
Example 1
cont’d
The contribution to X2 from the first cell is
Cells 2, 3, and 4 contribute .658, .274, and .108,
respectively, so X2 = .433 + .658 + .274 + .108 = 1.473.
A test with significance level .10 requires
, the number
in the 3 df row and .10 column of Appendix Table A.7.
This critical value is 6.251. Since 1.473 is not at least
6.251, H0 cannot be rejected even at this rather large level
of significance. The data is quite consistent with Mendel’s
laws.
22
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
Although we have developed the chi-squared test for
situations in which k > 2, it can also be used when k = 2.
The null hypothesis in this case can be stated as
H0: p1 = p10, since the relations p2 = 1 – p1 and p20 = 1 – p10
make the inclusion of p2 = p20 in H0 redundant.
The alternative hypothesis is Ha: p1 ≠ p10. These
hypotheses can also be tested using a two-tailed z test with
test statistic
23
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
Surprisingly, the two test procedures are completely
equivalent. This is because it can be shown that that
Z2 = X2 and
so that
if and only if (iff )
If the alternative hypothesis is either Ha: p1 > p10 or
Ha: p1 < p10, the chi-squared test cannot be used. One must
then revert to an upper- or lower-tailed z test.
As is the case with all test procedures, one must be careful
not to confuse statistical significance with practical
significance.
24
Goodness-of-Fit Tests When Category Probabilities Are Completely Specified
A computed X2 that exceeds
may be a result of a very
large sample size rather than any practical differences
between the hypothesized pi0’s and true pi’s.
Thus if p10 = p20 = p30 = , but the pi0’s true pi’s have
values .330, .340, and .330, a large value of X2 is sure to
arise with a sufficiently large n.
Before rejecting H0, the s should be examined to see
whether they suggest a model different from that of H0 from
a practical point of view.
25
P-Values for Chi-Squared Tests
26
P-Values for Chi-Squared Tests
The chi-squared tests in this section are all upper-tailed, so
we focus on this case.
Just as the P-value for an upper-tailed t test is the area
under the tv curve to the right of the calculated t, the
P-value for an upper-tailed chi-squared test is the area
under the curve to the right of the calculated X2.
Appendix Table A.7 provides limited P-value information
because only five upper-tail critical values are tabulated for
each different .
27
P-Values for Chi-Squared Tests
The fact that t curves were all centered at zero allowed us
to tabulate t-curve tail areas in a relatively compact way,
with the left margin giving values ranging from 0.0 to 4.0 on
the horizontal t scale and various columns displaying
corresponding uppertail areas for various df’s.
The rightward movement of chi-squared curves as df
increases necessitates a somewhat different type of
tabulation.
The left margin of Appendix Table A.11 displays various
upper-tail areas: .100, .095, .090, . . . , .005, and .001.
28
P-Values for Chi-Squared Tests
Each column of the table is for a different value of df, and
the entries are values on the horizontal chi-squared axis
that capture these corresponding tail areas.
For example, moving down
to tail area .085 and across
to the 4 df column, we see
that the area to the right
of 8.18 under the 4 df
chi-squared curve is .085
(see Figure 14.2).
A P-value for an upper-tailed chi-squared test
Figure 14.2
29
P-Values for Chi-Squared Tests
To capture this same upper-tail area under the 10 df curve,
we must go out to 16.54.
In the 4 df column, the top row shows that if the calculated
value of the chi-squared variable is smaller than 7.77, the
captured tail area (the P-value) exceeds .10.
Similarly, the bottom row in this column indicates that if the
calculated value exceeds 18.46, the tail area is smaller
than .001(P-value < .001).
30
χ2 When the Pi’s Are
Functions of Other Parameters
31
χ2 When the Pi’s Are Functions of Other Parameters
Sometimes the pi’s are hypothesized to depend on a
smaller number of parameters 1,…, m (m < k).
Then a specific hypothesis involving the i’s yields specific
pi0’s, which are then used in the X2 test.
32
Example 2
In a well-known genetics article (“The Progeny in
Generations F12 to F17 of a Cross Between a YellowWrinkled and a Green-Round Seeded Pea,” J. of Genetics,
1923: 255–331), the early statistician G. U. Yule analyzed
data resulting from crossing garden peas.
The dominant alleles in the experiment were Y = yellow
color and R = round shape, resulting in the double
dominant YR.
Yule examined 269 fourseed pods resulting from a dihybrid
cross and counted the number of YR seeds in each pod.
33
Example 2
cont’d
Letting X denote the number of YRs in a randomly selected
pod, possible X values are 0, 1, 2, 3, 4, which we identify
with cells 1, 2, 3, 4, and 5 of a rectangular table (so, e.g., a
pod with X = 4 yields an observed count in cell 5).
The hypothesis that the Mendelian laws are operative and
that genotypes of individual seeds within a pod are
independent of one another implies that X has a binomial
distribution with n = 4 and  = .
We thus wish to test H0: p1 = p1 = p10 ,…, p5 = p50, where
pi0 = P(i – 1 YRs among 4 seeds when H0 is true)
34
Example 2
pi0 =
cont’d
 i – 1(1 – )4 – (i – 1) i = 1, 2, 3, 4, 5;  =
Yule’s data and the computations are in Table 14.3, with
expected cell counts npi0 = 269pi0.
Observed and Expected Cell Counts for Example 2
Figure 14.3
35
Example 2
cont’d
Thus X2 = 3.823 + …. + .032 = 4.582.
Since
=
= 13.277, H0 is not rejected at level .01.
Appendix Table A.11 shows that because 4.582 < 7.77, the
P-value for the test exceeds .10.
H0 should not be rejected at any reasonable significance
level.
36
χ2 When the Underlying
Distribution Is Continuous
37
χ2 When the Underlying Distribution Is Continuous
We have so far assumed that the k categories are naturally
defined in the context of the experiment under
consideration.
The X2 test can also be used to test whether a sample
comes from a specific underlying continuous distribution.
Let X denote the variable being sampled and suppose the
hypothesized pdf of X is f0(x).
38
χ2 When the Underlying Distribution Is Continuous
As in the construction of a frequency distribution in Chapter
1, subdivide the measurement scale of X into k intervals
[a0, a1), [a1, a2),…, [ak – 1, ak), where the interval [ai –1, ai)
includes the value ai – 1 but not ai.
The cell probabilities specified by H0 are then
pi0 = P(ai –1  X < ai) =
f0(x) dx
The cells should be chosen so that npi0  5 for i = 1,…, k.
Often they are selected so that the npi0’s are equal.
39
Example 3
To see whether the time of onset of labor among expectant
mothers is uniformly distributed throughout a 24-hour day,
we can divide a day into k periods, each of length 24/k. The
null hypothesis states that f(x) is the uniform pdf on the
interval [0, 24], so that pi0 = 1/k.
The article “The Hour of Birth” (British J. of Preventive and
Social Medicine, 1953: 43–59) reports on 1186 onset times,
which were categorized into k = 24 1-hour intervals
beginning at midnight, resulting in cell counts of 52, 73, 89,
88, 68, 47, 58, 47, 48, 53, 47, 34, 21, 31, 40, 24, 37, 31, 47,
34, 36, 44, 78, and 59.
40
Example 3
Each expected cell count is 1186 
resulting value of X2 is 162.77.
cont’d
= 49.42, and the
Since
= 41.637, the computed value is highly
significant, and the null hypothesis is resoundingly rejected.
Generally speaking, it appears that labor is much more
likely to commence very late at night than during normal
waking hours.
41
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