The Bayley Scales of Infant Development yield scores on two indices - the Psychomotor Development Index (PDI) and the Mental Development Index (MDI) - that can be used to assess a child's level of functioning at approximately one year of age. Among normal healthy infants, both indices have a mean value of 100. As part of a study investigating the development and neurologic status of children who had undergone reparative heart surgery during the first three months of life, the Bayley Scales were administered to a sample of one-year-old infants born with congenital heart failure. The children had been randomized to one of two different treatment groups, known as "circulatory arrest" and "low-flow bypass".

The groups differed in the specific way in which the reparative surgery was performed. Unlike circulatory arrest, low-flow bypass maintains continuous circulation through the brain; although it is felt to be preferable by some physicians, it also has its own associated risk of brain injury.

The data for this study are in the "Heart" text file. PDI scores are saved under the variable name pdi, MDI scores under mdi, and indicators of treatment group under trtment. For this variable, 0 represents circulatory arrest and 1 is for low-flow bypass.

(a)Previously, you used a two-sample t-test to compare mean MDI score for infants assigned to the circulatory arrest and low-flow bypass. This analysis can also be performed using a simple linear regression model. Fit a linear regression model with MDI as the response variable and the indicator of treatment group as the explanatory variable. Write down the population model and the estimated model from R (be sure to define any variables you use).

(b)Who is more likely to have a higher MDI score, a child assigned to the circulatory arrest treatment group or one assigned to the low-flow bypass group? How much higher would the score be on average? (Note that, due to the way that treatment is coded, the slope for the treatment variable corresponds to a comparison of the low-flow bypass to the circulatory arrest procedure rather than the other way around.)

(c)Use your regression model to test if there is a treatment group difference in MDI score at the 5% significance level? Be sure to include the following:

-Explicitly define the parameter of interest. -State the null and alternative hypothesis. -State the level of the significance test. -State and check necessary assumptions. -State the form of the test statistic and its null distribution. -State the p-value and give the relevant R output. -State your conclusion in the context of the problem.

(d)Calculate the 95% confidence interval for the regression slope using R. Interpret this confidence interval.

(e)How does the confidence interval for the slope that you computed in part (d) compare with a 95% confidence interval for the difference in the mean MDI values for the two groups?

"Heart" text file is shown as below.

trtment pdi mdi 0 80 74 1 118 124 1 122 109 0 98 78 0 98 91 0 111 130 1 111 119 0 82 115 1 99 112 0 86 119 1 92 100 0 122 115 1 98 117 0 78 108 0 92 115 0 86 112 1 90 82 0 92 142 0 104 102 1 92 130 0 105 103 1 98 93 1 80 96 1 50 50 1 87 112 0 67 106 1 80 93 0 76 110 1 98 117 1 98 96 1 92 126 0 80 91 1 98 106 0 86 91 1 92 117 0 86 91 1 105 86 1 115 118 0 105 98 1 118 117 1 92 115 0 92 130 0 70 91 1 117 115 0 80 96 0 80 96 1 98 98 1 98 117 0 117 103 1 92 122 0 63 96 1 115 105 0 71 100 1 98 122 0 111 117 0 111 122 0 63 103 1 98 109 1 98 91 1 115 120 1 98 96 1 80 70 1 92 86 0 86 81 0 111 100 1 110 102 0 98 83 0 82 97 1 93 102 1 99 97 0 86 115 1 92 103 0 86 103 0 60 78 0 130 114 1 122 109 1 98 107 0 80 89 1 92 130 0 80 112 1 122 112 1 98 83 0 98 122 1 70 103 0 60 115 1 98 119 1 92 119 0 105 115 1 104 100 0 75 58 0 87 102 0 110 115 0 93 131 0 63 86 0 104 110 1 104 112 0 98 98 0 111 103 0 111 117 1 124 122 1 114 118 0 98 70 1 92 109 0 105 112 1 86 122 1 86 119 1 92 117 0 70 56 1 120 122 0 105 115 0 99 89 1 99 107 1 111 109 1 98 98 1 93 120 0 98 106 0 98 98 1 93 107 0 77 109 1 105 103 0 105 103 1 87 97 0 105 140 0 99 94 0 117 106 0 92 109 0 134 131 0 86 100 1 75 89 0 87 82 1 115 107 1 80 91 1 109 111 1 80 112 0 105 103 1 52 78 0 92 108 1 105 109 0 66 93 0 92 100 1 110 97 0 87 100 0 87 112

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