Handbook_of_statistical_analysis_using_SAS

The final question to address is whether or not mortality and calcium concentration differ in northern and southern towns. Because the distribution of mortality appears to be approximately normal, a t-test can be applied. Calcium concentration has a relatively high degree of skewness; thus, applying a Wilcoxon test or a t-test after a log transformation may be more sensible. The relevant SAS instructions are as follows.

data water; set water;

lhardnes=log(hardness); proc ttest;

class location;

var mortal hardness lhardnes; proc npar1way wilcoxon;

class location; var hardness;

run;

The short data step computes the (natural) log of hardness and stores it in the data set as the variable lhardnes. To use proc ttest, the variable that divides the data into two groups is specified in the class statement and the variable (or variables) whose means are to be compared are specified in the var statement. For a Wilcoxon test, the npar1way procedure is used with the wilcoxon option.

The results of the t-tests are shown in Display 2.13; those for the Wilcoxon tests in Display 2.14. The t-test for mortality gives very strong evidence for a difference in mortality in the two regions, with that in the North being considerably larger (the 95% confidence interval for the difference is 185.11, 328.47). Using a test that assumes equal variances in the two populations or one that does not make this assumption (Satterthwaite [1946]) makes little difference in this case. The t-test on the untransformed hardness variable also indicates a difference, with the mean hardness in the North being far less than amongst towns in the South. Notice here that the test for the equality of population variances (one of the assumptions of the t-test) suggests that the variances differ. In examining the results for the log-transformed variable, it is seen that the t-test still indicates a highly significant difference, but in this case the test for homogeneity is nonsignificant.

The result from the nonparametric Wilcoxon test (Display 2.14) once again indicates that the mean water hardness of towns in the North differs from that of towns in the South.

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The TTEST Procedure

Statistics

Display 2.13

The NPAR1WAY Procedure

Wilcoxon Scores (Rank Sums) for Variable Hardness

Classified by Variable location

Average scores were used for ties.

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Wilcoxon Two-Sample Test

Z includes a continuity correction of 0.5.

Display 2.14

Exercises

2.1Rerun proc univariate with the plot option for line printer plots.

2.2Generate box plots of mortality and water hardness by location (use proc boxplot).

2.3Use proc univariate to compare the distribution of water hardness to the log normal and exponential distributions.

2.4Produce histograms of both mortality and water hardness with, in each case, a kernel density estimate of the variable’s distribution superimposed.

2.5Produce separate perspective plots of the estimated bivariate densities of northern and southern towns.

2.6Reproduce the scatterplot in Display 2.10 with added linear regression fits of mortality and hardness for both northern and southern towns. Use different line types for the two regions.

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Chapter 3

Simple Inference for Categorical Data: From Sandflies to Organic Particulates in the Air

3.1 Description of Data

This chapter considers the analysis of categorical data. It begins by looking at tabulating raw data into cross-classifications (i.e. contingency tables) using the mortality and water hardness data from Chapter 2. It then examines six data sets in which the data are already tabulated; a description of each of these data sets is given below. The primary question of interest in each case involves assessing the relationship between pairs of categorical variables using the chi-square test or some suitable alternative.

The six cross-classified data sets to be examined in this chapter are as follows:

1.Sandflies (Table 128 in SDS). These data are given in Display 3.1 and show the number of male and female sandflies caught in light traps set 3 ft and 35 ft above the ground at a site in eastern Panama. The question of interest is: does the proportion of males and females caught at a particular height differ?

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Display 3.1

2.Acacia ants (Table 27 in SDS). These data, given in Display 3.2, record the results of an experiment with acacia ants. All but 28 trees of two species of acacia were cleared from an area in Central America, and the 28 trees were cleared of ants using insecticide. Sixteen colonies of a particular species of ant were obtained from other trees of species A. The colonies were placed roughly equidistant from the 28 trees and allowed to invade them. The question of interest is whether the invasion rate differs for the two species of acacia tree.

Display 3.2

3.Piston ring failures (Table 15 in SDS). These data are reproduced in Display 3.3 and show the number of failures of piston rings in each of three legs in each of four steam-driven compressors located in the same building. The compressors have identical design and are orientated in the same way. The question of interest is whether the pattern of the location of failures is different for different compressors.

4.Oral contraceptives. These data appear in Display 3.4 and arise from a study reported by Sartwell et al. (1969). The study was

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conducted in a number of hospitals in several large American cities. In those hospitals, all those married women identified as suffering from idiopathic thromboembolism (blood clots) over a 3-year period were individually matched with a suitable control, those being female patients discharged alive from the same hospital in the same 6-month time interval as the case. In addition, they were individually matched to cases on age, marital status, race, etc. Patients and controls were then asked about their use of oral contraceptives.

Display 3.3

Display 3.4

5.Oral lesions. These data appear in Display 3.5; they give the location of oral lesions obtained in house-to-house surveys in three geographic regions of rural India.

6.Particulates in the air. These data are given in Display 3.6; they arise from a study involving cases of bronchitis by level of organic particulates in the air and by age (Somes and O’Brien [1985]).

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Display 3.5 (Taken from Table 18.1 of the User Manual for Proc.-Stat Xact-4 for SAS Users, Cytel Software Corporation, Cambridge, MA, 2000.)

Display 3.6 (Reprinted by permission from the Encyclopedia of Statistical Sciences, John Wiley & Sons, Inc., New York, Copyright © 1982.)

3.2 Methods of Analysis

Contingency tables are one of the most common ways to summarize categorical data. Displays 3.1, 3.2, and 3.4 are examples of 2 × 2 contingency tables (although Display 3.4 has a quite different structure from Displays

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3.1 and 3.2 as explained later). Display 3.3 is an example of a 3 × 4 table and Display 3.5 an example of a 9 × 3 table with very sparse data. Display 3.6 is an example of a series of 2 × 2 tables involving the same two variables. For all such tables, interest generally lies in assessing whether or not there is an association between the row variable and the column variable that form the table. Most commonly, a chi-square test of independence is used to answer this question, although alternatives such as Fisher’s exact test or McNemar’s test may be needed when the sample size is small (Fisher’s test) or the data consists of matched samples (McNemar’s test). In addition, in 2 × 2 tables, it may be required to calculate a confidence interval for the difference in two population proportions. For a series of 2 × 2 tables, the Mantel-Haenszel test may be appropriate (see later). (Details of all the tests mentioned are given in Everitt [1992].)

3.3 Analysis Using SAS

3.3.1 Cross-Classifying Raw Data

We first demonstrate how raw data can be put into the form of a crossclassification using the data on mortality and water hardness from Chapter 2.

data water;

infile 'n:\handbook\datasets\water.dat';

input flag $ 1 Town $ 2-18 Mortal 19-22 Hardness 25-27; if flag='*' then location='north';

else location='south'; mortgrp=mortal > 1555; hardgrp=hardness > 39;

run;

proc freq data=water;

tables mortgrp*hardgrp / chisq; run;

The raw data are read into a SAS data set water, as described in Chapter 2. In this instance, two new variables are computed — mortgrp and hardgrp

— that dichotomise mortality and water hardness at their medians. Strictly speaking, the expression mortal > 1555 is a logical expression yielding the result “true” or “false,” but in SAS these are represented by the values 1 and 0, respectively.

Proc freq is used both to produce contingency tables and to analyse them. The tables statement defines the table to be produced and specifies

the analysis of it. The variables that form the rows and columns are joined

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with an asterisk (*); these may be numeric or character variables. Oneway frequency distributions are produced where variables are not joined

by asterisks. Several tables can be specified in a single tables statement. The options after the “/” specify the type of analysis. The chisq option

requests chi-square tests of independence and measures of association based on chi-square. The output is shown in Display 3.7. We leave commenting on the contents of this type of output until later.

The FREQ Procedure

Table of mortgrp by hardgrp

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Fisher's Exact Test

Sample Size = 61

Display 3.7

Now we move on to consider the six data sets that actually arise in the form of contingency tables. The freq procedure is again used to analyse such tables and compute tests and measures of association.

3.3.2 Sandflies

The data on sandflies in Display 3.1. can be read into a SAS data set with each cell as a separate observation and the rows and columns identified as follows:

data sandflies;

input sex $ height n; cards;

m 3 173 m 35 125 f 3 150

f 35 73

;

The rows are identified by a character variable sex with values m and f. The columns are identifed by the variable height with values 3 and 35.

The variable n contains the cell count. proc freq can then be used to analyse the table.

proc freq data=sandflies;

tables sex*height /chisq riskdiff; weight n;

run;

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