Question

A study was conducted to investigate the use of community-based treatment programs among Medicaid beneficiaries suffering from severe mental illness. The study involved assigning a sample of 311 patients to a prepaid medical plan and a sample of 310 patients to the traditional Medicaid program. After a specified period of time, the number of persons in each group who had visited a community crisis center in the previous three months was determined. Among the subjects assigned to the prepaid plan, 13 had visited a crisis center; among those receiving traditional Medicaid, 22 had visited a crisis center.

a. Using Stata, test the null hypothesis that the proportions are identical in the two populations, at the 0.10 significance level.

b. From the Stata output, what are your estimates of the two proportions?

c. Using the information from the Stata output, calculate the 95% confidence interval for the true difference between the proportions.

d. Use Stata to determine how much power you have to detect the observed difference at the 5% significance level.

Answer 1

A study was conducted to investigate the use of community-based treatment programs among Medicaid beneficiaries suffering from severe mental illness. The study involved assigning a sample of 311 patients to a prepaid medical plan and a sample of 310 patients to the traditional Medicaid program. After a specified period of time, the number of persons in each group who had visited a community crisis center in the previous three months was determined. Among the subjects assigned to the prepaid plan, 13 had visited a crisis center; among those receiving traditional Medicaid, 22 had visited a crisis center.

a. Using Stata, test the null hypothesis that the proportions are identical in the two populations, at the 0.10 significance level.

prtesti 311 13 310 22, level(90) count

Two-sample test of proportions                     x: Number of obs =      3

> 11

                                                   y: Number of obs =      3

> 10

----------------------------------------------------------------------------

> --

    Variable |       Mean   Std. Err.      z    P>|z|     [90% Conf. Interva

> l]

-------------+--------------------------------------------------------------

> --

           x |   .0418006   .0113485                       .023134    .06046

> 73

           y |   .0709677   .0145836                      .0469798    .09495

> 56

-------------+--------------------------------------------------------------

> --

        diff | -.0291671   .0184789                     -.0595622     .0012

> 28

             | under Ho:   .0185087    -1.58   0.115

----------------------------------------------------------------------------

> --

        diff = prop(x) - prop(y)                                  z = -1.57

> 59

    Ho: diff = 0

    Ha: diff < 0                 Ha: diff != 0                 Ha: diff > 0

Pr(Z < z) = 0.0575         Pr(|Z| > |z|) = 0.1151          Pr(Z > z) = 0.94

> 25

Calculated z= -1.57, P= 0.1151 which is > 0.10 level of significance.

There is not enough evidence to reject the proportions are identical in the two populations.

.

b. From the Stata output, what are your estimates of the two proportions?

p1=0.0418006  

p2=.0709677  

c. Using the information from the Stata output, calculate the 95% confidence interval for the true difference between the proportions.

       Proportion diff = -.0291671 standard error =   .0184789    

95% lower limit =-0.0291671 -1.96*0.0184789   =-0.0654

95% upper limit =-0.0291671 +1.96*0.0184789   =0.00705

d. Use Stata to determine how much power you have to detect the observed difference at the 5% significance level.

power twoproportions 0.0418 0.07097, test(chi2) n1(311) n2(310)

Estimated power for a two-sample proportions test

Pearson's chi-squared test

Ho: p2 = p1 versus Ha: p2 != p1

Study parameters:

        alpha =    0.0500

            N =       621

           N1 =       311

           N2 =       310

        N2/N1 =    0.9968

        delta =    0.0292 (difference)

           p1 =    0.0418

           p2 =    0.0710

Estimated power:

        power =    0.3505