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.

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`.

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

The cases are births and there are 1000 cases (ran dim on nc to get to this number).

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.

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?

boxplot(nc$weight~nc$habit)

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

set.seed(23)
weight_means500 <- rep(NA, 5000)

for(i in 1:5000){
   wsamp <- sample(nc$weight, 500)
   weight_means500[i] <- mean(wsamp)
   }

hist(weight_means500, breaks = 25)

There are more than 30 cases, the weight data resembles a normal curve, and cases are independent so conditions for inference are met.

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

H₀ = The average weights of babies born to smoking and non-smoking mothers are the same.

H_A = The average weights of babies born to smoking and non-smoking mothers are different.

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")

## 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.

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", 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 )

I don’t see any difference between the results of the ci and ht plots

On your own

Calculate a 95% confidence interval for the average length of pregnancies (weeks) and interpret it in context. Note that since you’re doing inference on a single population parameter, there is no explanatory variable, so you can omit the x variable from the function.

inference(y = nc$weeks, est = "mean", type = "ci", 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 )

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.

inference(y = nc$weeks, est = "mean", type = "ci", 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 )

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

inference(y = nc$weight, x = nc$mature, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical", 
          order = c("younger mom","mature mom"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_younger mom = 867, mean_younger mom = 7.0972, sd_younger mom = 1.4855
## n_mature mom = 133, mean_mature mom = 7.1256, sd_mature mom = 1.6591

## Observed difference between means (younger mom-mature mom) = -0.0283
## 
## H0: mu_younger mom - mu_mature mom = 0 
## HA: mu_younger mom - mu_mature mom != 0 
## Standard error = 0.152 
## Test statistic: Z =  -0.186 
## p-value =  0.8526

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.

I’ll make new dfs filtering by mature factors, then run summaries on each. I know I could just do max on younger and min on older, but I like to see more data.

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

ncagemat <- select(nc, mage:mature) 

younger <- filter(ncagemat, mature == "younger mom")
mature <-  filter(ncagemat, mature == "mature mom")

summary(younger)

##       mage               mature   
##  Min.   :13.00   mature mom :  0  
##  1st Qu.:21.00   younger mom:867  
##  Median :25.00                    
##  Mean   :25.44                    
##  3rd Qu.:30.00                    
##  Max.   :34.00

summary(mature)

##       mage               mature   
##  Min.   :35.00   mature mom :133  
##  1st Qu.:35.00   younger mom:  0  
##  Median :37.00                    
##  Mean   :37.18                    
##  3rd Qu.:38.00                    
##  Max.   :50.00

Younger moms are from age 13 - 34 while mature moms are from 35 - 50. 

While I understand that this is a way to differentiate periods in a woman's life where she is most likely to have a successful, healthy birth, I feel like the labels are both misleading (I believe that maturity is more closely associated with being "of birthing age" than being past it) and presumptive, since it provides an artificial determinant. It would make more sense to create factors based on the data.

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.

a) Do male children weigh the same as female children?

H~0~ = The average weights of male children is the same as for female children.

H~A~ = The average weights of male and female babies are different.

inference(y = nc$weight, x = nc$gender, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical", 
          order = c("male","female"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_male = 497, mean_male = 7.3015, sd_male = 1.5168
## n_female = 503, mean_female = 6.9029, sd_female = 1.4759

## Observed difference between means (male-female) = 0.3986
## 
## H0: mu_male - mu_female = 0 
## HA: mu_male - mu_female != 0 
## Standard error = 0.095 
## Test statistic: Z =  4.211 
## p-value =  0

The male mean is 7.3 while female mean is 6.9, a difference of 0.4. The SD for each gender is about 1.5, which seems pretty high and I’m guessing that is because premies are included so we’re going to have to first do the analysis on full term babies and then on premies.

b) Do full term male children weigh the same as full term female children?

H~0~ = The average weights of full term male children is the same as for full term female children.

H~A~ = The average weights of full term male and full term female babies are different.

ncterm <- select(nc, c(premie, weight, gender)) 

fterm <- filter(ncterm, premie == "full term")
premie <- filter(ncterm, premie == "premie")

inference(y = fterm$weight, x = fterm$gender, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical", 
          order = c("male","female"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_male = 415, mean_male = 7.7106, sd_male = 1.0538
## n_female = 431, mean_female = 7.2176, sd_female = 1.0403

## Observed difference between means (male-female) = 0.493
## 
## H0: mu_male - mu_female = 0 
## HA: mu_male - mu_female != 0 
## Standard error = 0.072 
## Test statistic: Z =  6.845 
## p-value =  0

Still very surprised and confused by this result.

Just for giggles I’m going to change the null value to .5 to make sure it’s working right.

inference(y = fterm$weight, x = fterm$gender, est = "mean", type = "ht", null = .5, 
          alternative = "twosided", method = "theoretical", 
          order = c("male","female"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_male = 415, mean_male = 7.7106, sd_male = 1.0538
## n_female = 431, mean_female = 7.2176, sd_female = 1.0403

## Observed difference between means (male-female) = 0.493
## 
## H0: mu_male - mu_female = 0.5 
## HA: mu_male - mu_female != 0.5 
## Standard error = 0.072 
## Test statistic: Z =  -0.098 
## p-value =  0.9222

Okay, good. So there’s basically no chance that they are the same but it’s almost certain that they are within half a pound of each other. This also means that the SD wasn’t just a function of premies being mixed with full term babies, but since I’d already said I’d wanted to look at that…

c) Do premie male children weigh the same as premie female children?

H~0~ = The average weights of premie male children is the same as for premie female children.

H~A~ = The average weights of premie male and premie female babies are different.

inference(y = premie$weight, x = premie$gender, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical", 
          order = c("male","female"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_male = 81, mean_male = 5.2072, sd_male = 1.7915
## n_female = 71, mean_female = 5.0386, sd_female = 2.1644

## Observed difference between means (male-female) = 0.1686
## 
## H0: mu_male - mu_female = 0 
## HA: mu_male - mu_female != 0 
## Standard error = 0.325 
## Test statistic: Z =  0.519 
## p-value =  0.604

The means are much closer, the SDs are larger, and there are no outliers so there is a better chance that there is no difference between males and females in terms of weight in premies.

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.