e-Appendix. tECHNICAL NOTES. Regression analysis of aggregate continuous data.

Contents

1)Theory

2)SAS syntax to conduct Regression analysis of aggregate continuous data

3)aggregate dataset

4)individual-level dataset

1)theory

Binary and count outcome. For binary and count data, regression analyses based on the frequency of the outcome for all combinations of a set of categorical predictors provide effect estimates and standard errors identical to those based on the original individual-level data. This is possible because with categorical data the variance is a function of the mean. However, for continuous data, errors are independent from the mean, which makes the estimation of the variance using aggregate data more challenging.

Continuous outcome. To perform linear regression analyses on a continuous aggregate outcome we only need two parameters: the mean and the standard deviation, within strata of a set of categorical predictors. The frequencies (counts) of the combinations of the predictors can be used as weights. The standard deviation of the raw (individual-level) data for all combinations of the predictors can be used to calculate the pooled variance that subsequently can be used to correct the standard errors of the estimated parameters using aggregate data.

In the following we use a linear regression model with one binary predictor to illustrate how we obtain parameter estimates using aggregate data that are equal to the estimated parameters using raw data. Let be a vector of dependent variable be the design matrix, be vector of parameters, and be vector of errors. It is well known that the parameters can be estimated by the equation . It is easy to show that where k is the number of and n-k is the number of .

To aggregate the individual data we use as outcome, as design, as weight matrix, and as parameters. It is well know that which can be shown that it is equal to . Thus the point estimates of an individual-level regression model and aggregate data regression model when the predictors are categorical are exactly the same.

When we aggregate the data we reduce the variability of the outcome measure. Since the point estimates of both individual and aggregate regression models are the same, it follows that the sum of squares of both models will be the same. Having the individual total sum of squares or pooled variance, calculated from thestratum-specific standard deviations, and aggregate model sum of squares allows us to estimate by and consequently obtain the correct standard error for the estimated parameters using aggregate data. Alternatively, we can estimate by using the pooled variance with aggregate data.The above results are true regardless of the number of categorical predictors and the inclusion of product terms in the models.Means and standard deviations based on small cell counts would still produce unbiased parameter estimates. Strata of the categorical covariates should include at least two observations.

2)SAS syntax to conduct Regression analysis of aggregate continuous data

PROCIMPORTOUT= WORK.WIC

DATAFILE= "H:\aggregate_data\wic.txt"

DBMS=TAB REPLACE;

GETNAMES=YES;

DATAROW=2;

RUN;

data wic; set wic;

label latecare='Late or no prenatal care';

wtgaink =wtgain *0.453592;

label wtgaink='Weight gain during pregnancy (kg)';

drop wtgain ;

proccontents;

run;

* Fitting regression model to individual data;

Title'Original model (model 2)';

procglmdata=wic;

class wic mracethn latecare;

model wtgaink = wic mracethn latecare / solutionclparm;

odsoutput ParameterEstimates=p;

run;

* aggregating data

calculating average and std for each combination of predictors;

procsortdata=wic; by wic mracethn latecare; run;

procmeansdata=wic noprint;

by wic mracethn latecare;

var wtgaink ;

outputout=test1(keep=wic mracethn latecare meany stdy N) mean(wtgaink)=meany std(wtgaink)=stdy N=N;

run;

* Fitting regression model to aggregated data;

procglmdata=test1;

odslistingclose;

class wic mracethn latecare;

model meany = wic mracethn latecare /solutionclparm;

weight N;

odsoutput OverallANOVA=anova;

odsoutput ParameterEstimates=p1;

run;

* Calculating pooled variance;

data test1; set test1;

n_1=N-1;

S_n_1=stdy**2*n_1;

run;

procmeansdata=test1 noprint;

var n_1 s_n_1;

outputout=o2 sum(n_1)=n_k sum(S_n_1)=S2_k;

run;

data o2; set o2; pooled_var=s2_k/n_k; call symput('p_v',pooled_var); run;

* the macro variable p_v contain pooled-variance.

* in the following we calculate error sum of squares and degress of freedom;

data anova; set anova;

if source='Error'thencall symput('errorms',ms);

run;

* the parameter estimate from aggregated data is used for

recalculating standard error and p-values using Z test;

data p1; set p1;

* we calculate inflation factor;

factor=sqrt(&p_v/&errorms);

standard_error=StdErr*factor;

z=estimate/standard_error;

p = 2*(1 - cdf('normal',abs(z)));

if p lt 0.0001then pvalue='<0.0001';

else pvalue=put(p,7.4);

if p=.then pvalue=' ';

lower_CL=estimate-1.96*standard_error;

upper_CL=estimate+1.96*standard_error;

run;

title'Aggregated model using pooled variance';

odslisting;

procprintdata=p1 noobslabel;

var parameter estimate standard_error lower_cl upper_cl z pvalue;

run;

3)aggregate dataset

Number of mothers, mean and standard deviation of weight gain during pregnancy, by strata of categorical covariates

WIC / MRACETHN / Late entry to prenatal care (LATECARE) / Weight gain, kg
(WTGAIN)
Receipt of WIC food / Race/ethnicity / N / Mean / (SD)
No / Hispanics / No / 105 / 14.2 / (5.6)
Yes / 112 / 13.2 / (5.6)
Non-Hispanic Whites / No / 307 / 15.5 / (5.5)
Yes / 291 / 15.7 / (6.1)
Non-Hispanic Blacks / No / 116 / 15.3 / (7.1)
Yes / 110 / 14.0 / (6.6)
Yes / Hispanics / No / 846 / 14.8 / (5.5)
Yes / 514 / 14.3 / (5.9)
Non-Hispanic Whites / No / 1292 / 16.2 / (5.9)
Yes / 846 / 16.1 / (6.3)
Non-Hispanic Blacks / No / 441 / 14.9 / (5.7)
Yes / 290 / 15.3 / (6.4)

Abbreviations: SD: Standard Deviation; Kg: Kilograms

4)individual-level dataset

WTGAINWIC MRACETHNLATECARE

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27Y20

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51Y11

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45Y30

20Y21

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40N30

26Y20

46Y11

34Y11

35N21

20Y10

33Y10

33Y10

28Y11

21Y11

42Y21

25Y21

26Y20

30N11

16N11

16Y20

75N21

37Y10

29N20

23Y21

37Y30

34N21

30N10

22Y11

43Y21

25Y11

68Y10

27N21

26Y10

18N10

23Y10

35Y10

19Y11

32N21

20Y31

28Y20

29Y21

40Y10

28Y11

22N31

20Y31

44Y21

34Y10

32N30

48Y10

21Y20

61Y10

46N31

43Y21

37Y20

47Y21

40Y11

13Y10

38Y20

56Y21

52Y11

35Y21

25Y10

37Y20

25Y11

18Y30

57Y20

26N11

24Y10

27N30

46Y10

37Y21

40Y20

28Y21

17Y21

20Y11

57Y21

21Y10

29Y11

12N11

52Y10

33Y21

18Y11

49Y11

20Y11

20Y10

19Y20

30Y20

20Y10

16Y11

57Y10

39Y30

46Y21

36Y21

17Y31

19N10

38N21

26N20

40Y10

15Y11

34Y10

16Y11

40N20

79N30

43Y30

35Y21

25Y11

45Y11

40Y11

20Y11

58Y30

32Y10

41Y10

33Y11

29Y10

21Y10

15Y11

19Y11

24N31

35Y10

43N11

31Y31

35Y20

41Y20

37Y10

7Y11

38Y20

29Y11

48Y21

36Y10

23Y10

23N21

22Y10

25N21

30Y21

31Y11

20Y10

57N21

41Y21

28N11

15Y11

95Y11

25Y20

46Y20

61Y10

50Y21

15N10

25Y11

55Y20

10N10

25N31

40Y11

58Y10

26Y10

51Y20

50Y11

22N31

8Y21

40Y10

18N11

25Y11

49Y11

63N11

24Y11

34Y20

35Y10

35Y21

45N21

62Y10

29Y31

50N20

33N11

40Y21

30Y10

40N21

39Y21

33N11

18Y11

30Y10

48Y11

36Y10

21Y11

15N21

50N21

34Y20

50N21

27Y10

25N10

42Y10

26N30

40Y30

27Y10

27Y21

27N31

58N21

61Y20

31Y10

47Y21

32N20

78Y10

81Y21

25Y20

62Y21

21Y21

25Y20

52N20

47Y10

31Y10

33Y30

24Y10

25Y10

25Y21

27Y11

38Y10

32N10

16Y11

18Y10

66Y21

33Y10

28Y10

47N21

27Y10

30Y31

18N11

33Y30

28Y10

30Y11

25Y11

38Y11

32N21

20N21

35Y31

34Y31

18N31

49Y10

16Y11

19N30

39N21

32Y10

30N11

28Y20

32Y11

20Y11

31N10

14Y21

20Y21

20Y10

25Y10

33Y20

61Y21

23N11

40N21

70Y20

44Y10

41Y21

30N10

35Y10

40Y31

22Y10

35N20

51Y30

18Y31

15Y10

28Y11

42Y20

36Y11

34Y10

30Y31

66N10

26Y10

32N20

41Y10

57N21

36Y20

29N20

32Y11

35Y10

26Y11

20Y11

29N11

15Y30

49Y30

38Y20

16Y11

34N21

30N10

28Y21

26N31

21N30

31Y31

34Y21

33N30

23Y20

21Y11

32Y11

30N31

37N21

38Y21

26N10

36Y31

29Y30

84Y31

24Y11

83Y10

64N21

26Y11

2N10

54Y10

65Y10

34Y20

47Y21

41Y10

35Y21

24Y21

24Y20

26Y11

36Y21

18Y31

28Y21

19Y20

21N11

25Y20

20Y11

40Y20

34Y21

47Y20

25Y20

44Y20

42Y20

49N20

33Y20

37Y10

39N20

35Y20

19Y20

31Y10

29Y20

40Y11

29N21

21Y11

26Y20

50Y10

47N10

32N20

39N20

20Y10

42N21

25Y21

47Y20

36Y30

22Y31

45N20

43Y30

21Y11

30Y10

28Y21

42Y31

28Y20

30Y31

20Y21

26N20

15Y30

42Y20

42Y20

26Y21

11Y11

70Y21

30Y30

25N21

20Y11

33Y21

22Y21

50N30

15N10

16Y20

17Y11

32N20

48Y20

20Y11

26Y10

27Y20

67Y21

43Y20

27Y30

17N30

35Y10

24Y20

46Y20

41Y20

36Y20

31Y20

28Y20

50Y10

30Y20

22Y10

40N21

56N20

13Y11

36Y10

30Y21

21Y11

15Y31

24N21

26Y20

11Y10

24Y10

32N21

30Y11

36Y20

31N20

25Y21

43Y21

30Y10

45Y20

46N10

33Y10

38Y21

34Y20

48Y20

28Y20

16Y10

40Y20

39N20

28Y10

41Y10

19Y11

26Y20

23Y10

56Y21

41Y20

39Y10

21Y31

23N31

44Y20

28Y21

16N10

28Y21

54Y10

25N20

34Y21

61N10

18N20

20Y20

57Y21

63Y21

36N21

53Y10

30Y10

33Y21

30N21

15Y10

23Y11

49Y21

34Y20

37N10

28Y20

48N20

27Y31

23Y31

32N20

33Y10

28Y20

38Y11

28Y10

26Y20

7N20

59Y20

42Y30

18Y20

60Y10

49Y10

16Y31

21Y20

35Y10

42Y20

30Y11

2N10

31Y20

32Y20

36Y20

26Y20

20Y10

14Y10

26Y10

41N21

34Y20

65Y10

18Y20

23Y21

27Y30

48N11

10Y11

27N30

26Y10

34Y10

42Y30

12Y11

40Y20

71Y30

27Y20

31Y10

24Y20

41Y30

36Y20

38N20

31N31

29Y21

44Y30

72Y11

53Y21

20Y20

32N20

39Y20

40Y20

45Y21

35Y20

39Y20

26N20

20Y11

23Y10

41Y10

14Y11

38Y10

35N10

28N30

26Y11

17Y11

18N20

43N31

38Y21

48Y10

44Y30

35N11

26N11

40Y31

42Y21

17N10

29Y21

47Y10

11Y11

39Y30

59Y31

48Y10

55Y30

30Y10

45N20

21Y21

17Y20

40Y20

45Y20

33Y10

29Y10

31Y11

42Y11

44Y11

12Y11

69Y30

57Y10

25N20

46Y30

29Y10

26Y10

25Y21

35Y10

42Y30

21Y11

44Y30

19Y11

56Y11

23Y11

16Y30

48Y11

24Y20

52N20

36Y10

22Y31

58Y20

31Y30

13Y20

19Y30

45Y10

23Y11

25Y11

34Y21

32Y20

18N20

32Y30

43Y11

20Y11

36Y21

34Y20

30Y11

47N31

24Y20

19Y31

45Y11

47Y30

59Y11

33Y20

20N31

53Y10

42Y21

34Y20

45Y21

41N20

25N20

35Y30

37N31

31Y10

40Y30

22Y30

43Y10

42Y10

27Y20

70Y10

30N31

32Y10

25Y10

32Y31

46Y11

25N31

35Y20

37Y31

35Y21

30Y11

31Y30

13N20

24Y10

22Y30

16Y20

29Y30

19Y30

49Y11

35Y31

32Y20

31Y10

24N20

20N21

53Y30

24Y10

27Y11

31Y11

34Y30

31Y10

32Y11

30Y11

24Y10

25Y10

20N11

43Y20

25Y11

18N30

25Y20

33Y20

33Y20

18Y10

22Y10

61Y10

22Y21

35Y11

21N11

33Y11

26Y30

49N20

24Y10

44Y11

21N21

12Y10

28Y10

35Y20

23Y11

32Y10

28Y10

67Y31

21Y10

41N21

26Y10

20Y10

51N20

30N11

67Y10

35N20

49Y30

29Y21

39Y31

28Y31

23N10

8Y11

58Y30

46Y31

21N11

27Y30

51Y31

40Y31

48Y11

35Y11

32Y11

40Y21

30Y11

20Y31

25Y10

57Y30

35Y31

44Y10

27Y21

20Y31

22Y30

40Y31

32Y31

23Y11

20Y10

21Y30

41Y10

21Y11

44Y11

43Y11

17Y11

45N30

30Y20

31Y20

32Y11

33N21

50Y30

18Y20

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