Example

Protein in Pregnancy

Data were collected through interest in whether, and if so, in what way the level of protein changes in expectant mothers throughout their pregnancy. Observations were taken on 19 healthy women. Each woman was at a different stage of pregnancy, gestation.

We have seen this example previously for parameter estimation assessing model fit.

\(\newline\) Data: \((y_i,x_i) \quad i=1,\dots,19\)

\(\newline\) Model: \(E(Y_i) = \alpha+\beta x_i\)

\(\newline\) Perform a hypothesis test to test \(H_0:\beta=0\).

We will use the R output to answer this question, however please make sure you are able to estimate regression coefficients by hand.

pregnancy<-read.csv("week6/Lecture11/PROTEIN.CSV",header=T)
fit1<-lm(formula = Protein ~ Gestation,data=pregnancy)
summary(fit1)

## 
## Call:
## lm(formula = Protein ~ Gestation, data = pregnancy)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16853 -0.08720 -0.01009  0.08578  0.20422 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.201738   0.083363   2.420    0.027 *  
## Gestation   0.022844   0.003295   6.934 2.42e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1151 on 17 degrees of freedom
## Multiple R-squared:  0.7388, Adjusted R-squared:  0.7234 
## F-statistic: 48.08 on 1 and 17 DF,  p-value: 2.416e-06

\(\newline\) The hypotheses being tested for the coefficient of \(\beta\) are:

\(\newline\) \(H_0: \beta = 0\)

\(\newline\) \(H_1: \beta \neq 0\)

\(\newline\) From the regression output

\[ \frac{\hat{\beta}}{\text{e.s.e}(\hat{\beta})}=\frac{0.022844}{0.003295}= 6.932929\sim t(n-p) \mbox{ under } H_0. \]

Since the p-value for gestation is \(<0.001\) (and hence \(<0.05\)) the null hypothesis is rejected and we conclude that there is a statistically significant relationship between protein and gestation. The gestational age is a useful predictor of the protein level.