DataM: HW Exercise 0316 3

The IQ scores and behavioral problem scores of children at age 5 were examined depending on whether or not their mothers had suffered an episode of post-natal depression. The main questions of interests were:
(1) Did the two groups of children have different IQ and/or behavioral problems?
(2) Was there any evidence of a relationship between IQ and behavioral problems?

Source: Dr. C. Kumar, Institute of Psychiatry, Lonon.

Load in the data set and check its structure

Load in the data set and named it dta.

dta <- read.table("../data/IQ_Beh.txt", header = T, row.names = 1)

Find the class of dta.

class(dta)

[1] "data.frame"

dta is a data frame.

Check the structure of dta.

str(dta)

'data.frame':   94 obs. of  3 variables:
 $ Dep: Factor w/ 2 levels "D","N": 2 2 2 2 1 2 2 2 2 2 ...
 $ IQ : int  103 124 124 104 96 92 124 99 92 116 ...
 $ BP : int  4 12 9 3 3 3 6 4 3 9 ...

dta is a data frame with 94 observations and 3 variables: Dep, IQ, and BP. Dep is a factorial variable. IQ and BP are numerical variable with integers.

Check the dimension of dta.

dim(dta)

[1] 94  3

dta has 94 rows and 3 columns.

Find the names of the columns in dta.

names(dta)

[1] "Dep" "IQ"  "BP"

Check if BP, a variable in dta, is a vector or not.

is.vector(dta$BP)

[1] TRUE

BP, a variable in dta, is a vector.

Take the $1^{st}$ row of dta, it is a slice of dta.

dta[1, ]

Take the $1^{st} - 3^{rd}$ elements of IQ, a variable in dta. It is a vector.

dta[1:3, "IQ"]

[1] 103 124 124

Sort dta in the ascending order of variable BP, and take the last 6 rows. In other words, this code takes the data of the 6 highest BP.

tail(dta[order(dta$BP), ])

Sort dta in the descending order of variable BP, and take the last 4 rows. In other words, this code takes the data of the 4 lowest BP.

tail(dta[order(-dta$BP), ], 4)

Data visualization

Draw the histogram of ˋˋdta$IQˋˋ with x-axis name, “IQ” and without the title.

with(dta, hist(IQ, xlab = "IQ", main = ""))

Draw the box plots of dta$BP in groups of dta$Dep with names of x-axis and y-axis.

boxplot(BP ~ Dep, data = dta, 
        xlab = "Depression", 
        ylab = "Behavior problem score")

Compare to participants in the group of non-depression, participant in the group of depression seemed to have higher levels of behavioral problems.

Draw the scatter plot of dta$IQ and dta$BP with different colors representing different groups of Dep. Set the point style and add a grid on the plot.

plot(IQ ~ BP, data = dta, pch = 20, col = dta$Dep, 
     xlab = "Behavior problem score", ylab = "IQ")
grid()

Draw the scatter plot of dta$BP and dta$IQ with point labels of Dep. Add the regression lines of y=BP, x=IQ with different types representing different groups of Dep.

plot(BP ~ IQ, data = dta, type = "n",
     ylab = "Behavior problem score", xlab = "IQ")
text(dta$IQ, dta$BP, labels = dta$Dep, cex = 0.5)
abline(lm(BP ~ IQ, data = dta, subset = Dep == "D"))
abline(lm(BP ~ IQ, data = dta, subset = Dep == "N"), lty = 2)

Compare to the group of non-depression, there were a more negative correlation (i.e., $r<0$) between IQ and behavioral problems in the group of depression.

Hypothesis testing

For question (a): Independent two sample t-test

Did the two groups of children have different IQ?

$H_0: \mu_{IQ(D)} = \mu_{IQ(N)}$
$H_1: \mu_{IQ(D)} \neq \mu_{IQ(N)}$

t.test(IQ ~ Dep, data = dta)


    Welch Two Sample t-test

data:  IQ by Dep
t = -1.6374, df = 15.53, p-value = 0.1216
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -26.926586   3.490299
sample estimates:
mean in group D mean in group N 
       101.0667        112.7848

Since $p > \alpha =.05$, we retain $H_0$. Two groups of children did not have significantly different IQ.

Did the two groups of children have different behavioral problems?

$H_0: \mu_{BP(D)} = \mu_{BP(N)}$
$H_1: \mu_{BP(D)} \neq \mu_{BP(N)}$

t.test(BP ~ Dep, data = dta)


    Welch Two Sample t-test

data:  BP by Dep
t = 1.4924, df = 17.14, p-value = 0.1538
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.6637017  3.8788916
sample estimates:
mean in group D mean in group N 
       7.000000        5.392405

Since $p > \alpha =.05$, we retain $H_0$. Two groups of children did not have significantly different behavioral problems.

For question (b): Correlation test and linar regression analysis

Was there any evidence of a relationship between IQ and behavioral problems?

Correlation test

$H_0: \phi_{IQ, BP} = 0$
$H_1: \phi_{IQ, BP} \neq 0$

cor.test(dta$IQ, dta$BP)


    Pearson's product-moment correlation

data:  dta$IQ and dta$BP
t = -3.8088, df = 92, p-value = 0.0002518
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.5319037 -0.1798969
sample estimates:
       cor 
-0.3690615

Since $p < \alpha =.05$, we reject $H_0$. IQ is significantly correlated with behavioral problems.

Regression analysis

$H_0: \beta_{IQ} = 0$
$H_1: \beta_{IQ} \neq 0$

summary(lm(BP ~ IQ, data = dta))


Call:
lm(formula = BP ~ IQ, data = dta)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.9828 -2.3564 -0.4111  2.1210  7.2399 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 13.18280    2.00180   6.585 2.76e-09 ***
IQ          -0.06792    0.01783  -3.809 0.000252 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.983 on 92 degrees of freedom
Multiple R-squared:  0.1362,    Adjusted R-squared:  0.1268 
F-statistic: 14.51 on 1 and 92 DF,  p-value: 0.0002518

Since $p < \alpha =.05$, we reject $H_0$. IQ is significantly associated with behavioral problems.