Score Test Statistic, Dean (1992)

The script begins by creating an internal dataset 'toxoplasmosis' via a DATA STEP and datalines, including variables for the number of successes (t), the total number of trials (m), and a 'rain' variable which is copied to 'z'. Then, the 'z' variable is standardized using PROC STDIZE. A PROC GLIMMIX is used to fit a generalized linear mixed model with a logit link and binomial distribution, modeling 't/m' as a function of 'z', 'z*z' and 'z*z*z'. Model predictions are exported for score statistic calculation. A subsequent DATA STEP calculates intermediate terms necessary for the test. PROC MEANS aggregates these terms, and a final DATA STEP calculates the Z-statistic of the score test and its associated p-value. Finally, PROC PRINT displays the results in formatted HTML output.

Data Analysis

Type : INTERNAL_CREATION

Data is created directly within the script via a DATA STEP and 'datalines'.

1 Code Block

DATA STEP Data

Explanation :
This DATA STEP block creates the 'toxoplasmosis' dataset by reading raw data (t, m, rain) directly from 'datalines'. A new variable 'z' is created as a copy of the 'rain' variable.

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1	DATA toxoplasmosis;
2	INPUT t m rain;
3	z = rain;
4	DATALINES;
5	2 4 1735
6	3 10 1936
7	1 5 2000
8	3 10 1973
9	2 2 1750
10	3 5 1800
11	2 8 1750
12	7 19 2077
13	3 6 1920
14	8 10 1800
15	7 24 2050
16	0 1 1830
17	15 30 1650
18	4 22 2200
19	0 1 2000
20	6 11 1770
21	0 1 1920
22	33 54 1770
23	4 9 2240
24	5 18 1620
25	2 12 1756
26	0 1 1650
27	8 11 2250
28	41 77 1796
29	24 51 1890
30	7 16 1871
31	46 82 2063
32	9 13 2100
33	23 43 1918
34	53 75 1834
35	8 13 1780
36	3 10 1900
37	1 6 1976
38	23 37 2292
39	;

2 Code Block

PROC STDIZE

Explanation :
This procedure standardizes the 'z' variable in the 'toxoplasmosis' dataset. The output dataset overwrites the original, ensuring that subsequent calculations use the standardized version of 'z'.

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1
2	PROC STDIZE
3	DATA=toxoplasmosis out=toxoplasmosis;
4	var z;
5
6	RUN;
7

3 Code Block

PROC GLIMMIX

Explanation :
PROC GLIMMIX is used to fit a generalized linear mixed model. The model specifies a binomial response (t/m), a logit link function, and includes 'z', 'z*z', and 'z*z*z' as predictors. The 's' option requests summary statistics. The 'output' clause creates a new dataset 'pdata' containing predicted probabilities ('pi') without random effects (noblup) and on the response scale (ilink).

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1	ods select none;
2	PROC GLIMMIX DATA=toxoplasmosis;
3	model t/m = z zz zz*z / link=logit dist=bin s;
4	OUTPUT out=pdata pred(noblup ilink) = pi;
5	RUN;
6	ods select all;

4 Code Block

DATA STEP

Explanation :
This DATA STEP reads the 'pdata' dataset (created by GLIMMIX) and calculates several intermediate variables ('pic', 'pipic', 'mpi', 't_mpi', 'pit_mpi', 'tpic', 'mm_1', 'aux') necessary for Dean's score test statistic formula. Only 'aux' and 'mm_1' are kept for subsequent steps.

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1	DATA pdata;
2	SET pdata;
3	pic = 1 - pi;
4	pipic = pi * pic;
5	mpi = m * pi;
6	t_mpi = t - mpi;
7	pit_mpi = pi * t_mpi;
8	tpic = t * pic;
9	mm_1 = m * (m-1);
10	aux = ( t_mpi*t_mpi + pit_mpi - tpic ) / pipic;
11	keep aux mm_1;
12	RUN;

5 Code Block

PROC MEANS

Explanation :
PROC MEANS is used here to calculate the sum of 'aux' and 'mm_1' variables across the entire 'pdata' dataset. The result is stored in a new dataset called 'new', and the 'noprint' option suppresses the display of PROC MEANS' default output.

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1	PROC MEANS DATA=pdata sum noprint;
2	var aux mm_1;
3	OUTPUT out=new sum=aux mm_1;
4	RUN;

6 Code Block

DATA STEP

Explanation :
This DATA STEP reads the 'new' dataset (containing the sums of 'aux' and 'mm_1') and calculates the Z-statistic of the score test as well as its p-value ('pval'). Formats and labels are applied to the variables for better presentation.

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1	DATA new;
2	SET new;
3	label Z = "GOF Test";
4	label PVal = "P-Value";
5	FORMAT Z 8.2 Pval pvalue6.;
6	z = aux / sqrt( 2*mm_1 );
7	pval = 1 - probnorm( z );
8	RUN;

7 Code Block

PROC PRINT

Explanation :
This block generates the final output. ODS HTML is enabled to direct output to an HTML file (or the SAS Studio environment). A title is set. PROC PRINT is used to display the 'z' (test statistic) and 'pval' (p-value) variables from the 'new' dataset. The 'noobs' option suppresses the observation number, and 'label' uses the defined labels for column headers. ODS HTML is then closed.

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1	ods html;
2	title "Score Test Statistic, Dean (1992)";
3	PROC PRINT DATA=new noobs label;
4	var z pval;
5	RUN;
6	ods html close;

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