Inference for numerical data

North Carolina births

In 2004, the state of North Carolina released a large data set containing information on births recorded in this state. This data set is useful to researchers studying the relation between habits and practices of expectant mothers and the birth of their children. We will work with a random sample of observations from this data set.

Exploratory analysis

Load the nc data set into our workspace.

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 3.5.1

load("more/nc.RData")
head(nc)

##   fage mage      mature weeks    premie visits marital gained weight
## 1   NA   13 younger mom    39 full term     10 married     38   7.63
## 2   NA   14 younger mom    42 full term     15 married     20   7.88
## 3   19   15 younger mom    37 full term     11 married     38   6.63
## 4   21   15 younger mom    41 full term      6 married     34   8.00
## 5   NA   15 younger mom    39 full term      9 married     27   6.38
## 6   NA   15 younger mom    38 full term     19 married     22   5.38
##   lowbirthweight gender     habit  whitemom
## 1        not low   male nonsmoker not white
## 2        not low   male nonsmoker not white
## 3        not low female nonsmoker     white
## 4        not low   male nonsmoker     white
## 5        not low female nonsmoker not white
## 6            low   male nonsmoker not white

We have observations on 13 different variables, some categorical and some numerical. The meaning of each variable is as follows.

variable	description
`fage`	father’s age in years.
`mage`	mother’s age in years.
`mature`	maturity status of mother.
`weeks`	length of pregnancy in weeks.
`premie`	whether the birth was classified as premature (premie) or full-term.
`visits`	number of hospital visits during pregnancy.
`marital`	whether mother is `married` or `not married` at birth.
`gained`	weight gained by mother during pregnancy in pounds.
`weight`	weight of the baby at birth in pounds.
`lowbirthweight`	whether baby was classified as low birthweight (`low`) or not (`not low`).
`gender`	gender of the baby, `female` or `male`.
`habit`	status of the mother as a `nonsmoker` or a `smoker`.
`whitemom`	whether mom is `white` or `not white`.

What are the cases in this data set? How many cases are there in our sample?

There are 13 cases.

summary(nc)

##       fage            mage            mature        weeks      
##  Min.   :14.00   Min.   :13   mature mom :133   Min.   :20.00  
##  1st Qu.:25.00   1st Qu.:22   younger mom:867   1st Qu.:37.00  
##  Median :30.00   Median :27                     Median :39.00  
##  Mean   :30.26   Mean   :27                     Mean   :38.33  
##  3rd Qu.:35.00   3rd Qu.:32                     3rd Qu.:40.00  
##  Max.   :55.00   Max.   :50                     Max.   :45.00  
##  NA's   :171                                    NA's   :2      
##        premie        visits            marital        gained     
##  full term:846   Min.   : 0.0   married    :386   Min.   : 0.00  
##  premie   :152   1st Qu.:10.0   not married:613   1st Qu.:20.00  
##  NA's     :  2   Median :12.0   NA's       :  1   Median :30.00  
##                  Mean   :12.1                     Mean   :30.33  
##                  3rd Qu.:15.0                     3rd Qu.:38.00  
##                  Max.   :30.0                     Max.   :85.00  
##                  NA's   :9                        NA's   :27     
##      weight       lowbirthweight    gender          habit    
##  Min.   : 1.000   low    :111    female:503   nonsmoker:873  
##  1st Qu.: 6.380   not low:889    male  :497   smoker   :126  
##  Median : 7.310                               NA's     :  1  
##  Mean   : 7.101                                              
##  3rd Qu.: 8.060                                              
##  Max.   :11.750                                              
##                                                              
##       whitemom  
##  not white:284  
##  white    :714  
##  NA's     :  2  
##                 
##                 
##                 
##

As you review the variable summaries, consider which variables are categorical and which are numerical. For numerical variables, are there outliers? If you aren’t sure or want to take a closer look at the data, make a graph.

str(nc)

## 'data.frame':    1000 obs. of  13 variables:
##  $ fage          : int  NA NA 19 21 NA NA 18 17 NA 20 ...
##  $ mage          : int  13 14 15 15 15 15 15 15 16 16 ...
##  $ mature        : Factor w/ 2 levels "mature mom","younger mom": 2 2 2 2 2 2 2 2 2 2 ...
##  $ weeks         : int  39 42 37 41 39 38 37 35 38 37 ...
##  $ premie        : Factor w/ 2 levels "full term","premie": 1 1 1 1 1 1 1 2 1 1 ...
##  $ visits        : int  10 15 11 6 9 19 12 5 9 13 ...
##  $ marital       : Factor w/ 2 levels "married","not married": 1 1 1 1 1 1 1 1 1 1 ...
##  $ gained        : int  38 20 38 34 27 22 76 15 NA 52 ...
##  $ weight        : num  7.63 7.88 6.63 8 6.38 5.38 8.44 4.69 8.81 6.94 ...
##  $ lowbirthweight: Factor w/ 2 levels "low","not low": 2 2 2 2 2 1 2 1 2 2 ...
##  $ gender        : Factor w/ 2 levels "female","male": 2 2 1 2 1 2 2 2 2 1 ...
##  $ habit         : Factor w/ 2 levels "nonsmoker","smoker": 1 1 1 1 1 1 1 1 1 1 ...
##  $ whitemom      : Factor w/ 2 levels "not white","white": 1 1 2 2 1 1 1 1 2 2 ...

The numeric variables are fage, mage, weeks, visits, gained, and weight. There are plenty of outliers in terms of length of pregnancy and weight of newborn, as can be seen from the boxplots below.

p <- ggplot(nc, aes(x=1, y=weeks))
p + geom_boxplot()

## Warning: Removed 2 rows containing non-finite values (stat_boxplot).

p <- ggplot(nc, aes(x=1, y=weight))
p + geom_boxplot()

Consider the possible relationship between a mother’s smoking habit and the weight of her baby. Plotting the data is a useful first step because it helps us quickly visualize trends, identify strong associations, and develop research questions.

Make a side-by-side boxplot of habit and weight. What does the plot highlight about the relationship between these two variables?

p <- ggplot(nc, aes(x=habit, y=weight))
p + geom_boxplot()

The box plots show how the medians of the two distributions compare, but we can also compare the means of the distributions using the following function to split the weight variable into the habit groups, then take the mean of each using the mean function.

by(nc$weight, nc$habit, mean)

## nc$habit: nonsmoker
## [1] 7.144273
## -------------------------------------------------------- 
## nc$habit: smoker
## [1] 6.82873

There is an observed difference, but is this difference statistically significant? In order to answer this question we will conduct a hypothesis test .

Inference

Check if the conditions necessary for inference are satisfied. Note that you will need to obtain sample sizes to check the conditions. You can compute the group size using the same by command above but replacing mean with length.

by(nc$weight, nc$habit, length)

## nc$habit: nonsmoker
## [1] 873
## -------------------------------------------------------- 
## nc$habit: smoker
## [1] 126

. The individual observations must be independent. A random sample from less than 10% of the population ensures the observations are independent. In experiments, we generally require that subjects are randomized into groups.

. Other conditions focus on sample size and skew. For example, if the sample size is too small, the skew too strong, or extreme outliers are present, then the normal model for the sample mean will fail.

Write the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different.

mean.nonsmoker <- round(mean(nc$weight[nc$habit == "nonsmoker"], na.rm = T),3)
mean.smoker <- round(mean(nc$weight[nc$habit == "smoker"], na.rm = T),3)
print(paste0("newborn weight of non-smoking mother: ", mean.nonsmoker))

## [1] "newborn weight of non-smoking mother: 7.144"

print(paste0("newborn weight of smoking mother: ", mean.smoker))

## [1] "newborn weight of smoking mother: 6.829"

mean.diff <- mean.nonsmoker - mean.smoker 
print(paste0("difference in means: ", mean.diff))

## [1] "difference in means: 0.315"

var.nonsmoker <- round(var(nc$weight[nc$habit == "nonsmoker"], na.rm=T),3)
var.smoker <- round(var(nc$weight[nc$habit == "smoker"], na.rm=T),3)
se <- round(sqrt((var.nonsmoker/ 873) + (var.smoker/126)),3)
t.score <- round(abs(qt(0.025, 872)),3)
print(paste0("estimated critical t-score: ", t.score))

## [1] "estimated critical t-score: 1.963"

print(paste0("standard error: ", se))

## [1] "standard error: 0.134"

low.bound <- round((mean.diff - t.score * se), 3)
high.bound <- round((mean.diff + t.score * se), 3)
print(paste0("95% confidence interval: ", low.bound, ", ", high.bound))

## [1] "95% confidence interval: 0.052, 0.578"

As the confidence interval is always positive, we can say with confidence the difference in newborn is NOT due to chance.

Next, we introduce a new function, inference, that we will use for conducting hypothesis tests and constructing confidence intervals.

inference(y = nc$weight, x = nc$habit, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")

## Warning: package 'BHH2' was built under R version 3.5.1

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862

## Observed difference between means (nonsmoker-smoker) = 0.3155
## 
## H0: mu_nonsmoker - mu_smoker = 0 
## HA: mu_nonsmoker - mu_smoker != 0 
## Standard error = 0.134 
## Test statistic: Z =  2.359 
## p-value =  0.0184

Change the type argument to "ci" to construct and record a confidence interval for the difference between the weights of babies born to smoking and non-smoking mothers.

By default the function reports an interval for (\(\mu_{nonsmoker} - \mu_{smoker}\)) . We can easily change this order by using the order argument:

inference(y = nc$weight, x = nc$habit, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical", 
          order = c("smoker","nonsmoker"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187

## Observed difference between means (smoker-nonsmoker) = -0.3155
## 
## Standard error = 0.1338 
## 95 % Confidence interval = ( -0.5777 , -0.0534 )

On your own

1

Calculate a 95% confidence interval for the average length of pregnancies (weeks) and interpret it in context.

mean.weeks <- round(mean(nc$weeks, na.rm=T),3)
print(paste0("Mean number of weeks of pregnancy: ", mean.weeks))

## [1] "Mean number of weeks of pregnancy: 38.335"

sd.weeks <- round(sd(nc$weeks, na.rm=T),3)
print(paste0("Standard deviation of weeks of pregnancy: ", sd.weeks))

## [1] "Standard deviation of weeks of pregnancy: 2.932"

conflevel <- 0.95
z.score <- abs(qnorm((1-conflevel)/2))
se.weeks <- round(z.score * (sd.weeks / (sqrt(length(nc$weeks)))),3)
low.bound.weeks <- mean.weeks - se.weeks
high.bound.weeks <- mean.weeks + se.weeks
print(paste0("95% confidence interval of avg. weeks of pregnancy: [", low.bound.weeks, ", ", high.bound.weeks, "]"))

## [1] "95% confidence interval of avg. weeks of pregnancy: [38.153, 38.517]"

2

Calculate a new confidence interval for the same parameter at the 90% confidence level. You can change the confidence level by adding a new argument to the function: conflevel = 0.90.

conflevel <- 0.90
z.score <- abs(qnorm((1-conflevel)/2))
se.weeks <- round(z.score * (sd.weeks / (sqrt(length(nc$weeks)))),3)
low.bound.weeks <- mean.weeks - se.weeks
high.bound.weeks <- mean.weeks + se.weeks
print(paste0("90% confidence interval of avg. weeks of pregnancy: [", low.bound.weeks, ", ", high.bound.weeks, "]"))

## [1] "90% confidence interval of avg. weeks of pregnancy: [38.182, 38.488]"

3

Conduct a hypothesis test evaluating whether the average weight gained by younger mothers is different than the average weight gained by mature mothers.

The null hypothesis states that the weight gain of young and mature mothers is equivalent.

mean.fage <- round(mean(nc$fage, na.rm = T),3)
print(paste0("Mean age of all mothers: ", mean.fage))

## [1] "Mean age of all mothers: 30.256"

mean.gained.young <- round(mean(nc$gained[nc$fage < mean.fage], na.rm = T),3) 
mean.gained.mature <- round(mean(nc$gained[nc$fage > mean.fage], na.rm = T),3) 
print(paste0("Mean weight gain by young mothers: ", mean.gained.young))

## [1] "Mean weight gain by young mothers: 31.282"

print(paste0("Mean weight gain by mature mothers: ", mean.gained.mature))

## [1] "Mean weight gain by mature mothers: 30.2"

by(nc$fage > mean.fage, nc$fage < mean.fage, length)

## nc$fage < mean.fage: FALSE
## [1] 404
## -------------------------------------------------------- 
## nc$fage < mean.fage: TRUE
## [1] 425

mean.gained.diff <- mean.gained.young - mean.gained.mature
var.young <- round(var(nc$gained[nc$fage < mean.fage], na.rm=T),3)
var.mature <- round(var(nc$gained[nc$fage > mean.fage], na.rm=T),3)
se <- round(sqrt((var.young/ 425) + (var.mature/404)),3)
t.score <- round(abs(qt(0.025, 425)),3)
print(paste0("estimated critical t-score: ", t.score))

## [1] "estimated critical t-score: 1.966"

print(paste0("standard error: ", se))

## [1] "standard error: 0.979"

low.bound <- round((mean.diff - t.score * se), 3)
high.bound <- round((mean.diff + t.score * se), 3)
print(paste0("95% confidence interval: ", low.bound, ", ", high.bound))

## [1] "95% confidence interval: -1.61, 2.24"

As 0 can be found in the 95% confidence interval, we cannot reject the null hypothesis. There is not sufficient evidence that the weight gain differs between young and mature mothers.

4

Now, a non-inference task: Determine the age cutoff for younger and mature mothers. Use a method of your choice, and explain how your method works.

sp <- ggplot(nc, aes(x=fage, y=gained))
sp + geom_point() + stat_smooth(method=lm, se=F)

## Warning: Removed 190 rows containing non-finite values (stat_smooth).

## Warning: Removed 190 rows containing missing values (geom_point).

In the hypothesis test, I used the mean age of 30 years old and 3 months. Viewing the scatterplot, there is no basis for determining a better break-off point, when age suddenly has an impact on weight gain.

5

Pick a pair of numerical and categorical variables and come up with a research question evaluating the relationship between these variables. Formulate the question in a way that it can be answered using a hypothesis test and/or a confidence interval. Answer your question using the inference function, report the statistical results, and also provide an explanation in plain language.

I would perform a student to determine if race has an impact on the number of hospital visits during pregnancy.

My hypothesis test is that

Average visits of white mothers - Average visits of non-white mothers = 0

inference(y = nc$visits, x = nc$whitemom, est = "mean", type = "ci", null = 0, 
          alternative = "onesided", method = "theoretical", 
          order = c("white","not white"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_white = 710, mean_white = 12.3014, sd_white = 3.7701
## n_not white = 279, mean_not white = 11.6272, sd_not white = 4.3644

## Observed difference between means (white-not white) = 0.6742
## 
## Standard error = 0.2971 
## 95 % Confidence interval = ( 0.0918 , 1.2565 )

The 95 % Confidence interval is ( 0.0918 , 1.2565 ). We have sufficient evidence to conclude that white women did go to the doctor more, but not much more than one-visit on average.