Homework 2 STAT 308

Problem 1.

• Part a.) In this case, \(p = <2e-16\), which is below \(\alpha = 0.5\). From this, we can reject the null hypothesis. There is sufficient evidence to reject the claim that \(\beta_1=0\). This implies that the predictor gestational age is a significant predictor of head circumference.

lowbirthweight <- read.csv("C:/Users/Kajal/Downloads/lowbirthwt.csv")
headcirc_gestageLM<- lm(headcirc ~ gestage, data=lowbirthweight)
summary(headcirc_gestageLM)

## 
## Call:
## lm(formula = headcirc ~ gestage, data = lowbirthweight)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5358 -0.8760 -0.1458  0.9041  6.9041 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.91426    1.82915    2.14   0.0348 *  
## gestage      0.78005    0.06307   12.37   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.59 on 98 degrees of freedom
## Multiple R-squared:  0.6095, Adjusted R-squared:  0.6055 
## F-statistic: 152.9 on 1 and 98 DF,  p-value: < 2.2e-16

• Part b.) \(SSR = 386.87\), \(SSE = 247.88\) and \(MSE = \frac{SSE}{n-2} = 2.529388\)

anova(headcirc_gestageLM)

## Analysis of Variance Table
## 
## Response: headcirc
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## gestage    1 386.87  386.87  152.95 < 2.2e-16 ***
## Residuals 98 247.88    2.53                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Part c.) Based on the R-squared value of 0.6094799, we can say that 60.95% of variation in head circumference is explained by the gestational age of the baby.

summary(headcirc_gestageLM)$r.squared 

## [1] 0.6094799

Problem 2.

• Part a.) In this case, \(p = <2e-16\), which is below \(\alpha = 0.5\). From this, we can reject the null hypothesis. There is sufficient evidence to reject the claim that \(\beta_1=0\). This implies that the predictor birth weight is a significant predictor of head circumference.

headcirc_birthwtLM<- lm(headcirc ~ birthwt, data=lowbirthweight)
summary(headcirc_birthwtLM)

## 
## Call:
## lm(formula = headcirc ~ birthwt, data = lowbirthweight)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.1622 -0.9399 -0.3071  0.5471 10.0398 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.822e+01  6.447e-01   28.26   <2e-16 ***
## birthwt     7.492e-03  5.699e-04   13.15   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.531 on 98 degrees of freedom
## Multiple R-squared:  0.6381, Adjusted R-squared:  0.6344 
## F-statistic: 172.8 on 1 and 98 DF,  p-value: < 2.2e-16

• Part b.) \(SSR = 405.06\), \(SSE = 229.69\) and \(MSE = \frac{SSE}{n-2} = 2.343776\)

anova(headcirc_birthwtLM)

## Analysis of Variance Table
## 
## Response: headcirc
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## birthwt    1 405.06  405.06  172.82 < 2.2e-16 ***
## Residuals 98 229.69    2.34                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Part c.) Based on the R-squared value of 0.6381409, we can say that 63.81% of variation in head circumference is explained by the birth weight of the baby.

summary(headcirc_birthwtLM)$r.squared 

## [1] 0.6381409

Problem 3.

• Part a.) In this case, \(p = <2e-16\), which is below \(\alpha = 0.5\). From this, we can reject the null hypothesis. There is sufficient evidence to reject the claim that \(\beta_1=0\). This implies that the predictor length is a significant predictor of head circumference.

headcirc_lengthLM<- lm(headcirc ~ length, data=lowbirthweight)
summary(headcirc_lengthLM)

## 
## Call:
## lm(formula = headcirc ~ length, data = lowbirthweight)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.0143 -1.0383 -0.2883  0.6013  8.9644 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.84406    1.85829   4.221 5.44e-05 ***
## length       0.50532    0.05024  10.059  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.785 on 98 degrees of freedom
## Multiple R-squared:  0.508,  Adjusted R-squared:  0.503 
## F-statistic: 101.2 on 1 and 98 DF,  p-value: < 2.2e-16

• Part b.) \(SSR = 322.45\), \(SSE = 312.30\) and \(MSE = \frac{SSE}{n-2} = 3.186735\)

anova(headcirc_lengthLM)

## Analysis of Variance Table
## 
## Response: headcirc
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## length     1 322.45  322.45  101.18 < 2.2e-16 ***
## Residuals 98 312.30    3.19                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Part c.) Based on the R-squared value of 0.5079886, we can say that 50.8% of variation in head circumference is explained by the length of the baby.

summary(headcirc_lengthLM)$r.squared 

## [1] 0.5079886

Problem 4.

Based on the highest R-Squared value, we can say that birth weight has the strongest linear association with head circumference. The higher R-Squared value implies that birth weight is the better predictor for head circumference.

Problem 5.

multLM<-lm(headcirc ~ gestage + birthwt, data=lowbirthweight)
summary(multLM)

## 
## Call:
## lm(formula = headcirc ~ gestage + birthwt, data = lowbirthweight)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0350 -0.7271 -0.0765  0.3472  8.5402 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 8.3080154  1.5789429   5.262 8.54e-07 ***
## gestage     0.4487328  0.0672460   6.673 1.56e-09 ***
## birthwt     0.0047123  0.0006312   7.466 3.60e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.274 on 97 degrees of freedom
## Multiple R-squared:  0.752,  Adjusted R-squared:  0.7469 
## F-statistic: 147.1 on 2 and 97 DF,  p-value: < 2.2e-16

anova(multLM)

## Analysis of Variance Table
## 
## Response: headcirc
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## gestage    1 386.87  386.87 238.378 < 2.2e-16 ***
## birthwt    1  90.46   90.46  55.739 3.597e-11 ***
## Residuals 97 157.42    1.62                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Part a.)
  • \(\hat\beta_0\) = 8.3080154. This implies that this is the expected mean value of headcircumference when all X=0.
  • \(\hat\beta_{ga}\) = 0.4487328. This implies that this is the expected mean value of headcircumference when all other X=0 besides gestational age.
  • \(\hat\beta_{bw}\) = 0.0047123. This implies that this is the expected mean value of headcircumference when all other X=0 besides birth weight.
• Part b.)
  • \(SSR\)=477.33
  • \(SSE\)=157.42
  • \(R^2\)=0.752

SSRfull<-(386.87+90.46)
SSRfull

## [1] 477.33

• Part c.)
  • 1- \(R^2\) = 1-0.752 = 0.248
  • Therefore, 24.8% of the total variation in head circumference remains “unexplained” in this analysis. It depends whether or not this is a large amount. In my opinion, that does seem like a lot for the model to not explain, but there are plenty of other predictors we did not add into the mix to make the model better.
• Part d.)
  • F Statistic = MSR/MSE = 147.3241
  • MSR = SSR/df = 477.33/2, MSE = 1.62
  • \(H_a\): \(\hat\beta_{ga}\) \(\neq\) \(\hat\beta_{bw}\) \(\neq\) 0

(477.33/2)/1.62

## [1] 147.3241

With a \(p = <2.2e-16\), which is below \(\alpha = 0.5\), we can reject the null hypothesis. There is sufficient evidence to reject the claim that \(\hat\beta_{ga}\) = \(\hat\beta_{bw}\) = 0. This implies that the predictors gestational age and birth weight are significant predictors of head circumference.

• Part e.) Based on the confidence intervals below, we see that a 95% C.I. for…
  • gestational age = (0.315268189, 0.582197507)
  • birth weight = (0.003459568, 0.005964999)
  • Since neither interval contains the null value, 0, there is a statistically significant difference between the groups.

confint(multLM, level=0.95)

##                   2.5 %       97.5 %
## (Intercept) 5.174250734 11.441780042
## gestage     0.315268189  0.582197507
## birthwt     0.003459568  0.005964999

Problem 6.

multLM2<-lm(headcirc ~ gestage + birthwt + length, data=lowbirthweight)
summary(multLM2)

## 
## Call:
## lm(formula = headcirc ~ gestage + birthwt + length, data = lowbirthweight)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0359 -0.7278 -0.0755  0.3469  8.5403 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  8.3200783  1.9555657   4.255 4.87e-05 ***
## gestage      0.4489698  0.0712246   6.304 8.83e-09 ***
## birthwt      0.0047183  0.0008521   5.537 2.67e-07 ***
## length      -0.0006928  0.0656161  -0.011    0.992    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.281 on 96 degrees of freedom
## Multiple R-squared:  0.752,  Adjusted R-squared:  0.7442 
## F-statistic: 97.03 on 3 and 96 DF,  p-value: < 2.2e-16

anova(multLM2)

## Analysis of Variance Table
## 
## Response: headcirc
##           Df Sum Sq Mean Sq  F value    Pr(>F)    
## gestage    1 386.87  386.87 235.9203 < 2.2e-16 ***
## birthwt    1  90.46   90.46  55.1642 4.536e-11 ***
## length     1   0.00    0.00   0.0001    0.9916    
## Residuals 96 157.42    1.64                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Part a.)
  • \(\hat\beta_0\) = 8.3200783. This implies that this is the expected mean value of headcircumference when all X=0.
  • \(\hat\beta_{ga}\) = 0.4489698. This implies that this is the expected mean value of headcircumference when all other X=0 besides gestational age.
  • \(\hat\beta_{bw}\) = 0.0047183. This implies that this is the expected mean value of headcircumference when all other X=0 besides birth weight.
  • \(\hat\beta_{l}\) = -0.0006928. This implies that this is the expected mean value of headcircumference when all other X=0 besides length.
• Part b.)
  • \(SSR\)=477.33
  • \(SSE\)=157.42
  • \(R^2\)=0.752

SSRfull<-(386.87+90.46+0.0)
SSRfull

## [1] 477.33

• Part c.)
  • 1- \(R^2\) = 1-0.752 = 0.248
  • Therefore, 24.8% of the total variation in head circumference remains “unexplained” in this analysis. It depends whether or not this is a large amount. In my opinion, that does seem like a lot for the model to not explain, but there are plenty of other predictors we did not add into the mix to make the model better.
• Part d.)
  • F Statistic = MSR/MSE = 97.02
  • MSR = SSR/df = 477.33/3, MSE = 1.64
  • \(H_a\): \(\hat\beta_{ga}\) \(\neq\) \(\hat\beta_{bw}\) \(\neq\) \(\hat\beta_{l}\) $$0

(477.33/3)/1.64

## [1] 97.01829

With a \(p = <2.2e-16\), which is below \(\alpha = 0.5\), we can reject the null hypothesis. There is sufficient evidence to reject the claim that \(\hat\beta_{ga}\) = \(\hat\beta_{bw}\) = \(\hat\beta_{l}\) = 0. This implies that the predictors gestational age, birth weight, and length are significant predictors of head circumference.

• Part e.) Based on the confidence intervals below, we see that a 95% C.I. for…
  • gestational age = (0.30759008, 0.590349620)
  • birth weight = (0.00302685, 0.006409729)
  • length = (-0.13093973, 0.129554084)
  • Since neither gestational age or birth weight contains the null value, 0, these predictors are statistically significant predictors for this model. Since length does contain the null value, 0, this is not a statistically significant predictor of head circumference for this model.

confint(multLM2, level=0.95)

##                   2.5 %       97.5 %
## (Intercept)  4.43831099 12.201845545
## gestage      0.30759008  0.590349620
## birthwt      0.00302685  0.006409729
## length      -0.13093973  0.129554084