 |
In Week 7 we saw how to compare two population proportions, 
and 
. In this section we consider proportions for more than two 0-1 populations.
If we have K such populations and we have random samples of size 
from the populations, then typically we want to test whether whether the
true population proportions are some hypothesized values 
. The most common example is whether all proportions are the same.
If the null hypothesis is true, then we would expect to get 
1's (that is `successes') in the ith sample. If we let 
denote the actual number of 1's in the ith sample, then we could
measure how far the observed data is from what we expect if the null
hypothesis is true by test statistic
A large (small) value of this statistic is evidence against (not against)
the null hypothesis.
We reject the null hypothesis if 
.
| One
Categorical Variable
|
A slightly different situation to the previous section is if we have a
random sample of n objects each of which can fall into exactly one of
K `categories' (for example, roll a die 60 times; each time the die
can be one of 6 values) and for the ith category we observe
occurrences among the n objects. Now we let 
represent the hypothesized proportion of objects in the whole population of
objects that fall into the ith category. Now we expect 
occurrences of category i and we can measure the distance between
observed and expected results by the same 
statistic as above. The only difference in the procedure is that the degrees
of freedom is now K-1 rather than K.
| Goodness
of Fit Testing
|
An example of the one category situation is when we have a random sample 
of size n from a continuous population and we want to test whether
the population has a particular distribution (such as normal). In this case
we could divide the range of the data into K intervals and count how
many of the X's fall into each interval. For example, if the
hypothesis is that the X's come from a uniform distribution on the
interval [0,1], we could divide [0,1] into the 10 intervals [0,.1), [.1,.2),
and so on and then count how many X's are in each interval. Again we
call the observed counts 
.
From the hypothesized distribution, we can calculate how many X's
should be in each interval (for the uniform example, 10% of the X's
should fall in each interval). Again we have
where
is the probability that an X falls in the ith interval.
In some cases, we need to estimate the parameters of the hypothesized
distribution. In testing for normality for example, in order to find the
's, we need to know the mean and variance of the population. If we use 
and 
as estimates of the true mean and variance, then we must further reduce the
degrees of freedom of the
statistic by 2 (one for each estimated parameter).
EXAMPLE:\ To see if there is a
seasonal effect for homicide, 1361 crimes were classified into the four
seasons, where 334 of them happened in spring, 372 in summer, 327 in Fall
and 328 in winter. Do we have enough evidence to show that the crime
frequencies are different for different seasons? Let
, 
be the proportions of crimes for the four seasons, respectively.
; 
at least one inequality exists.
= 1361*.25 = 340.25,
.
- Chi-squared est statistics = 4.034 with d.f.=4-1=3.
- The rejection region is
.
- Fail to reject
and conclude that there is not enough evidence to show that there is a
seasonal effect on the crime rate.
| Inference
for Two-Way Tables
We now turn to the case where we have objects that can be categorized in
two ways (such as gender and political party).
 |
| Descriptive
Tables
|
A two-way table with r rows and c columns contains 
sample counts. Let 
denote the number of observations in the ith row and the jth
column, 
, 
. The general form of the data table giving the sample counts is as
follows:
In this table 
are the column totals and 
are the row totals. Thus,
If the total sample size is n, then
For most
tables, a better understanding of the information contained in the table
is obtained by examining the column proportions/ which are
defined as the jth column proportion 
. The resulting entries in each column form the conditional
distribution/ of the row variable given that value of the column
variable. Note that the sum of the entries in each column should be 1
(making exception only for round off error). Tables giving entries as
proportions of row totals are also useful. Which description to use
depends upon the particular set of data being analyzed and what
questions are of interest.
| Models
and Hypotheses
|
The test procedure for
count data is sufficiently general so that it is valid for different
assumptions regarding the data. The one assumption that must remain
stringent however is that each experimental unit be counted only once in
the data table.
The following table is a summary of the population proportions where
a single SRS is taken from a single population and each observation is
classified into one cell of an
table.
The marginal proportions in this table are the sums of the
proportions in the rows and columns. Here the 
are the row sums and the 
are the column sums. The marginal proportions are easily interpreted as
probabilities. Each
is the probability that a randomly selected member of the population
falls in the ith row category. Similarly, each
is the probability that a randomly selected member of the population
falls in the jth column category.
| First
Model for
Tables
|
A SRS of size n is drawn from a population. Each individual in
the sample is classified according to two categorical variables. The
probabilities for the row classification are
and the probabilities for the column classification are
.
The null hypothesis is the the row and column classifications are
independent; that is, there is no relationship between the row and
column classifications. Letting 
denote the probability of an observation being classified in row i
and in column j, the null hypothesis is
The alternative hypothesis is the the row and column classifications
are dependent; that is, the row and column classifications are related
in some way. We write this alternative as
The second model is a natural extension of the comparison of two
proportions we studied in Section 9.2. That is, the c populations
are independently sampled and the number of possible outcomes in each
population is r where 
.
| Second
Model for
Tables
|
For each of c populations, independent SRSs of sizes 
are drawn. Each individual in a sample is classified according to a
categorical outcome variable with r possible values. For the jth
population the probability that an individual will fall into category i
is 
.
The null hypothesis is that the distributions of the outcome variable
are the same in all c populations. Letting
denote the proportion of population j in category i, the
null hypothesis is
The alternative hypothesis is 
: at least one of the equalities in 
does not hold.
The samples sizes from each of the populations are the column totals
in the sample count table. Call these sample sizes
. In the first model, the
are random variables. The total samples size n is set by the
researcher, and the column sums are known only after the data are
analyzed. For the second model, the column sums are the sample sizes
selected at the design phase of the research. The null hypothesis in
both models says that there is no relationship between the column
variable and the row variable. Although the hypothesis is expressed
differently, the test of the hypothesis in each case is the same.
| Expected
Counts
|
The statistic that tests
in
tables compares the sample counts with expected/ counts that
are calculated under the assumption that
is true. The expected count in the ijth cell of the table is
denoted by 
. For an
table, the expected counts/ are calculated from the marginal
totals in the samples count table using the formula
| Significance
Tests
|
To test
, that there is no relationship between the row and column
classifications, a statistic called the chi-square statistic/
is used. This statistic compares the sample counts with their expected
values. Specifically, we take the difference between the sample count
and its expected count, square these values, and divide by the expected
count, then sum over all entries. That is, to compare the sample and
expected counts we use a statistic 
, called the chi-square statistic. It is calculated from the following
formula:
where observed/ represents the sample counts, and expected/
represents the expected counts, and the sum is over all
entries in the sample or expected count tables.
To test
, we need a distribution to compare
to, under the assumptions that
is true. This leads us to the chi-squared distribution. The
distribution is described by a single parameter, its degrees of freedom.
Furthermore, the
distribution is skewed to the right.
The data for an
table can be obtained by random sampling as described by either of the
two models previously discussed.
The null hypothesis to be tested is that the row and column
classifications are independent (first model) or that the row
classification proportions for the c populations are all equal
(second model). The alternative hypothesis is that the null hypothesis
is not true.
The test statistic is the
statistic
If
is true, the statistic
has approximately a
distribution with (r - 1)(c - 1) degrees of freedom.
The p-value for the test is 
where
is a random variable having the 
distribution. The approximation is based on having a large sample. The
sample is judged large enough if the average of the expected counts is 5
or more, and the smallest expected count is 1 or more.
| Summary
|
EXAMPLE: Each of 250 job
applicants at a large firm was classified in two ways: (1) whether or
not they got a job offer; and (2) their ethnic group.
Do the data indicate that receiving a job offer is independent of the
ethnicity of the applicant? We obtain the following output from
Stataquest:
| col
row| 1 2 3 | Total
-----------+-------------------------+-------
1 | 24 13 18 | 55
2 | 124 39 32 | 195
-----------+-------------------------+-------
Total| 148 52 50 | 250
Pearson chi2(2) = 8.8688 Pr = 0.012
 | 
Receiving a job offer is independent of ethnicity
| 
There is a relationship between receiving a job offer and ethnicity
| Test Statistic:
= 8.869
| p-value: p-value = 0.012
| The
is rejected at 0.05 level. |
| | | |
|
| | | | | | | | | | | | | |
|
| Computer
Lab
|
Applicable StataQuest Commands:
Summaries 
Tables
Two-way (cross-tabulation)
| Concept
Lab
|
| Ch 19: Chi-square Goodness of Fit Test |
|
| | | | | | | | | |
