Handbook_of_statistical_analysis_using_SAS

region are associated. Here, the chi-square test is unsuitable because of the very sparse nature of the data.

The FREQ Procedure

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WARNING: 93% of the cells have expected counts less than 5. Chi-Square may not be a valid test.

The FREQ Procedure

Statistics for Table of site by region

Fisher's Exact Test

Sample Size = 27

Display 3.12

3.3.7 Particulates and Bronchitis

The final data set to be analysed in this chapter, the bronchitis data in Display 3.6, involves 2 × 2 tables for bronchitis and level of organic particulates for three age groups. The data could be collapsed over age and the aggregate 2 × 2 table analysed as described previously. However, the potential dangers of this procedure are well-documented (see, for

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example, Everitt [1992]). In particular, such pooling of contingency tables can generate an association when in the separate tables there is none. A more appropriate test in this situation is the Mantel-Haenszel test. For a series of k 2 × 2 tables, the test statistic for testing the hypothesis of no association is:

where ai, bi, ci, di represent the counts in the four cells of the ith table and ni is the total number of observations in the ith table. Under the null hypothesis of independence in all tables, this statistic has a chi-squared distribution with a single degree of freedom.

The data can be read in and analysed using the following SAS code:

data bronchitis;

;

proc freq data=bronchitis order=data;

Tables agegrp*level*bronch / cmh noprint; weight n;

run;

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The tables statement specifies a three-way tabulation with agegrp

defining the strata. The cmh option requests the Cochran-Mantel-Haenszel statistics and the noprint option suppresses the tables. The results are

shown in Display 3.13. There is no evidence of an association between level of organic particulates and suffering from bronchitis. The P-value associated with the test statistic is 0.64 and the assumed common odds ratio calculated as:

takes the value 1.13 with a confidence interval of 0.67, 1.93. (Since the Mantel-Haenszel test will only give sensible results if the association between the two variables is both the same size and same direction in each 2 × 2 table, it is generally sensible to look at the results of the Breslow-Day test for homogeneity of odds ratios given in Display 3.13. Here there is no evidence against homogeneity. The Breslow-Day test is described in Agresti [1996]).

The FREQ Procedure

Summary Statistics for level by bronch

Controlling for agegrp

Cochran-Mantel-Haenszel Statistics (Based on Table Scores)

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Breslow-Day Test for

Homogeneity of the Odds Ratios

Total Sample Size = 1462

Display 3.13

Exercises

3.1For the oral contraceptives data, construct a confidence interval for the difference in the proportion of women suffering blood clots who used oral contraceptives and the corresponding proportion for women not suffering blood clots.

3.2For the piston ring data, the “residuals” used in the text were simply observed frequency minus expected under independence. Those are not satisfactory for a number of reasons, as discussed in Everitt (1992). More suitable residuals are r and radj given by:

Calculate both for the piston ring data and compare what each of the three types have to say about the data.

3.3In the data given in Display 3.5, the frequencies for the Keral and Andhra regions are identical. Reanalyse the data after simply summing the frequencies for those two regions and reducing the number of columns of the table by one.

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Chapter 4

Multiple Regression:

Determinants of Crime

Rate in the United States

4.1 Description of Data

The data set of interest in this chapter is shown in Display 4.1 and consists of crime rates for 47 states in the United States, along with the values of 13 explanatory variables possibly associated with crime. (The data were originally given in Vandaele [1978] and also appear in Table 134 of SDS.)

A full description of the 14 variables in these data is as follows:

RCrime rate: the number of offences known to the police per 1,000,000 population

Age Age distribution: the number of males aged 14 to 24 years per 1000 of total state population

SBinary variable distinguishing southern states (S = 1) from the rest

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MNumber of males per 1000 females

NState population size in hundred thousands

WWealth, as measured by the median value of transferable goods and assets or family income (unit 10 dollars)

XIncome inequality: the number of families per 1000 earning below one half of the median income

The main question of interest about these data concerns how the crime rate depends on the other variables listed. The central method of analysis will be multiple regression.

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Display 4.1

4.2 The Multiple Regression Model

The multiple regression model has the general form:

where yi is the value of a continuous response variable for observation i, and x1i, x2i, …, xpi are the values of p explanatory variables for the same observation. The term i is the residual or error for individual i and represents the deviation of the observed value of the response for this individual from that expected by the model. The regression coefficients, β 0, β 1, …, β p, are generally estimated by least-squares.

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Significance tests for the regression coefficients can be derived by assuming that the residual terms are normally distributed with zero mean and constant variance σ 2. The estimated regression coefficient corresponding to a particular explanatory variable gives the change in the response variable associated with a unit change in the explanatory variable, conditional on all other explanatory variables remaining constant.

For n observations of the response and explanatory variables, the regression model can be written concisely as:

known constraints, the first column containing a series of ones corresponding to the term β 0 in Eq. (4.1); and the remaining columns containing values of the explanatory variables. The elements of the vectorββ are the regression coefficients β 0, β 1, …, β p, and those of the vector , the residual

terms 1, 2, …, n.

The regression coefficients can be estimated by least-squares, resulting

The variances and covariances of the resulting estimates can be found from

where s2 is defined below.

The variation in the response variable can be partitioned into a part due to regression on the explanatory variables and a residual. These can be arranged in an analysis of variance table as follows:

Note:DF: degrees of freedom, SS: sum of squares, MS: mean square.

The residual mean square s2 gives an estimate of σ 2, and the F-statistic is a test that β 1, β 2, …, β p are all zero.

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A measure of the fit of the model is provided by the multiple correlation coefficient, R, defined as the correlation between the observed values of the response variable and the values predicted by the model; that is

The value of R2 gives the proportion of the variability of the response variable accounted for by the explanatory variables.

For complete details of multiple regression, see, for example, Rawlings (1988).

4.3 Analysis Using SAS

Assuming that the data are available as an ASCII file uscrime.dat, they can be read into SAS for analysis using the following instructions:

data uscrime;

infile 'uscrime.dat' expandtabs;

input R Age S Ed Ex0 Ex1 LF M N NW U1 U2 W X; run;

Before undertaking a formal regression analysis of these data, it may be helpful to examine them graphically using a scatterplot matrix. This is essentially a grid of scatterplots for each pair of variables. Such a display is often useful in assessing the general relationships between the variables, in identifying possible outliers, and in highlighting potential multicollinearity problems amongst the explanatory variables (i.e., one explanatory variable being essentially predictable from the remainder). Although this is only available routinely in the SAS/INSIGHT module, for those with access to SAS/IML, we include a macro, listed in Appendix A, which can be used to produce a scatterplot matrix. The macro is invoked as follows:

%include 'scattmat.sas'; %scattmat(uscrime,R--X);

This assumes that the macro is stored in the file ‘scattmat.sas’ in the current directory. Otherwise, the full pathname is needed. The macro is called with the %scattmat statement and two parameters are passed: the name of the SAS data set and the list of variables to be included in the scatterplot matrix. The result is shown in Display 4.2.

The individual relationships of crime rate to each of the explanatory variables shown in the first column of this plot do not appear to be

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