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.

load("more/nc.RData")

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

Each case represents one birth. There are 1,000 cases in our sample.

As a first step in the analysis, we should consider summaries of the data. This can be done using the summary command:

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.

variable description
fage father’s age in years.

numerical mage | mother’s age in years. numerical 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.

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.

  1. Make a side-by-side boxplot of habit and weight. What does the plot highlight about the relationship between these two variables?
boxdata <- data.frame(nc$habit, nc$weight)
boxplot(boxdata)

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

  1. 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.

Conditions: Sample observations are independent Being that each baby only has one mother, and can only be born once, yes, this condition is met. Sample size is large (n >= 30)
Our sample sizes are 873 and 126, so this condition is met. Population distribution is not strongly skewed.
Weight is normally distributed. Habit is a categorical value, so I’m not sure how to determine whether this variable meets this condition.

by(nc$weight, nc$habit, length)
## nc$habit: nonsmoker
## [1] 873
## -------------------------------------------------------- 
## nc$habit: smoker
## [1] 126
hist(nc$weight)

plot(nc$habit)

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

\[ H_o: \mu :smoking mom baby weight = \mu :nonsmoking baby weight \\ H_a: \mu: smoking baby weight \neq \mu: nonsmoking baby weight \]

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 3.3.3
## 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

Let’s pause for a moment to go through the arguments of this custom function. The first argument is y, which is the response variable that we are interested in: nc$weight. The second argument is the explanatory variable, x, which is the variable that splits the data into two groups, smokers and non-smokers: nc$habit. The third argument, est, is the parameter we’re interested in: "mean" (other options are "median", or "proportion".) Next we decide on the type of inference we want: a hypothesis test ("ht") or a confidence interval ("ci"). When performing a hypothesis test, we also need to supply the null value, which in this case is 0, since the null hypothesis sets the two population means equal to each other. The alternative hypothesis can be "less", "greater", or "twosided". Lastly, the method of inference can be "theoretical" or "simulation" based.

  1. 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.
inference(y = nc$weight, x = nc$habit, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical")
## 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
## 
## Standard error = 0.1338 
## 95 % Confidence interval = ( 0.0534 , 0.5777 )

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

inference(y = nc$weeks, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical")
## Single mean 
## Summary statistics:
## mean = 38.3347 ;  sd = 2.9316 ;  n = 998 
## Standard error = 0.0928 
## 95 % Confidence interval = ( 38.1528 , 38.5165 )
#this block below not working - need to debug. Getting NA's
#weeks_mean95 <- mean(nc$weeks)
#se <- sd(nc$weeks) / sqrt(998)
#lower <- weeks_mean95 - 1.96 * se
#upper <- weeks_mean95 + 1.96 * se
#c(lower, upper)

#do the work manually until the above is figured out
library(psych)
## Warning: package 'psych' was built under R version 3.3.3

describe(nc$weeks)
##    vars   n  mean   sd median trimmed  mad min max range  skew kurtosis
## X1    1 998 38.33 2.93     39   38.66 1.48  20  45    25 -2.01     7.66
##      se
## X1 0.09
se <- 2.93 / sqrt(998)
lower <- 38.33 - 1.96 * se
upper <- 38.33 + 1.96 * se
c(lower, upper)
## [1] 38.14821 38.51179

We are 95% confident that our population mean lies between (38.14821, 38.51179).

inference(y = nc$weeks, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical", conflevel = 0.90)
## Single mean 
## Summary statistics:

## mean = 38.3347 ;  sd = 2.9316 ;  n = 998 
## Standard error = 0.0928 
## 90 % Confidence interval = ( 38.182 , 38.4873 )
#this code block below not working - need to debug. Getting NA's
#weeks_mean90 <- mean(nc$weeks)
#se <- sd(nc$weeks) / sqrt(998)
#lower <- weeks_mean90 - 1.645 * se
#upper <- weeks_mean90 + 1.645 * se
#c(lower, upper)

#do the work manually until the above is figured out
se <- 2.93 / sqrt(998)
lower <- 38.33 - 1.645 * se
upper <- 38.33 + 1.645 * se
c(lower, upper)
## [1] 38.17743 38.48257

We are 90% confident that our population mean lies between (38.17743, 38.48257).

\[ H_o: \mu :young mother weight gained = \mu :mature mother weight gained \\ H_a: \mu: young mother weight gained \neq \mu: mature mother weight gained \]

inference(y = nc$gained, x = nc$mature, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_mature mom = 129, mean_mature mom = 28.7907, sd_mature mom = 13.4824
## n_younger mom = 844, mean_younger mom = 30.5604, sd_younger mom = 14.3469
## Observed difference between means (mature mom-younger mom) = -1.7697
## 
## H0: mu_mature mom - mu_younger mom = 0 
## HA: mu_mature mom - mu_younger mom != 0 
## Standard error = 1.286 
## Test statistic: Z =  -1.376 
## p-value =  0.1686

by(nc$gained, nc$mature, describe)
## nc$mature: mature mom
##    vars   n  mean    sd median trimmed   mad min max range skew kurtosis
## X1    1 129 28.79 13.48     28   28.17 11.86   0  70    70 0.49     0.38
##      se
## X1 1.19
## -------------------------------------------------------- 
## nc$mature: younger mom
##    vars   n  mean    sd median trimmed   mad min max range skew kurtosis
## X1    1 844 30.56 14.35     30   30.03 13.34   0  85    85 0.45     0.77
##      se
## X1 0.49
#mature
se <- 13.48 / sqrt(129)
lower <- 28.79 - 1.96 * se
upper <- 28.79 + 1.96 * se
c(lower, upper)
## [1] 26.46378 31.11622
#young
se <- 14.35 / sqrt(844)
lower <- 30.56 - 1.96 * se
upper <- 30.56 + 1.96 * se
c(lower, upper)
## [1] 29.59186 31.52814

We are 95% confident that the population mean for weight gained by mature mothers is between (26.46378, 31.11622). We are 95% confident that the population mean for weight gained by young mothers is between (29.59186, 31.52814). The average weight gained by a mature mother in this sample is 28.79 pounds, while the average weight gained by a young mother in this sample is 30.56. Unfortunately, our p value is pretty high at 0.1686, so we are not going to reject the null hypothesis.

We can determine the age cutoff for younger and mature mothers using a single line of code that we used up above. It is as follows:

by(nc$mage, nc$mature, describe)
## nc$mature: mature mom
##    vars   n  mean   sd median trimmed  mad min max range skew kurtosis
## X1    1 133 37.18 2.43     37   36.79 1.48  35  50    15 1.98     5.85
##      se
## X1 0.21
## -------------------------------------------------------- 
## nc$mature: younger mom
##    vars   n  mean   sd median trimmed  mad min max range skew kurtosis
## X1    1 867 25.44 5.03     25   25.44 5.93  13  34    21 0.01    -1.03
##      se
## X1 0.17

We can see here that the youngest age of a mature mother is 35, while the oldest age of a mature mother is 50. The youngest age for a younger mother is 13, while the oldest age for a younger mother is 34. So it seems 34/35 is the cutoff for young vs. mature.

I would like to research mother’s age against whether or not they are white, to see if there is a statistically significant difference between race and age.

\[ H_o: \mu :white-mom-age = \mu :not-white-mom-age \\ H_a: \mu: white-mom-age \neq \mu: non-white-mom-age \]

inference(y = nc$mage, x = nc$whitemom, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_not white = 284, mean_not white = 25.331, sd_not white = 6.435
## n_white = 714, mean_white = 27.6499, sd_white = 5.9898
## Observed difference between means (not white-white) = -2.3189
## 
## H0: mu_not white - mu_white = 0 
## HA: mu_not white - mu_white != 0 
## Standard error = 0.443 
## Test statistic: Z =  -5.237 
## p-value =  0

With a p-value of 0, we can confidently reject the null hypothesis in favor of the alternative. There is a statistically significant difference in the age of mothers when comparing those that are white and those that are not. Non-white mothers are significantly younger than white mothers.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Mine Çetinkaya-Rundel from a lab written by the faculty and TAs of UCLA Statistics.