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?

We can check the number of rows in the sample using the nrow function. Each row is a case, and a case is a birth in North Carolina.
```
nrow(nc)
```
```
[1] 1000
```
There are 1000 cases.

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?

Below is a side-by-side boxplot of weights of newborns as a function of mother’s smoking habits. We can see that the weight is slightly higher for babies with a nonsmoker mother, but there is more variability for the nonsmoker babies. Without a statistical test, I won’t be able to say the difference between the two is significant.
```
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.

The three conditions that must be satisfied are independence, randomization, and normally distributed. We can see the numbers of samples for both nonsmoker and smoker with the following R-code:
```
by(nc$weight, nc$habit, length)
```
```
nc$habit: nonsmoker
[1] 873
-------------------------------------------------------- 
nc$habit: smoker
[1] 126
```
We can see that there are over 30 samples in both groups, so we can assume normality. We can also assume that 999 subjects is less than 10% of all the births, so independence is also satisfied. It is also safe to assume that the selection was random. Therefore, the conditions for inference are satisfied.
Write the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different.

$H_{0}: \mu_{NS} = \mu_{S}$

$H_{a}: \mu_{NS} \ne \mu_{S}$

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.

The confidence interval is displayed below:

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

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", 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 )

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", 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 )

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 and alternative hypothesis can be seen below:

$H_{0}: \mu_{MM} = \mu_{YM}$

$H_{a}: \mu_{MM} \ne \mu_{YM}$

where $MM$ is mature mother and $YM$ is young mother. We can use the inference function again:

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

If we assume a 95% confidence interval, we can see that the p-value is larger than 0.05. Therefore, we fail to reject the null hypothesis, and there is not sufficient evidence to say that average weight gained is different between mature and younger moms.

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.

We can use the by command again:
```
by(nc$mage,nc$mature,summary)
```
```
nc$mature: mature mom
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  35.00   35.00   37.00   37.18   38.00   50.00 
-------------------------------------------------------- 
nc$mature: younger mom
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  13.00   21.00   25.00   25.44   30.00   34.00 
```
From here we can see that the range for mature moms is 35-50, and the range for young moms is 13-34. Theregore the cutoff is for young moms is 34 years old.
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.

We could conduct a hypothesis test on whether or not the weight gained by mothers during pregnancy is dependent on race of the mother (white or not). The null and alternative hypothesis would look like this:

$H_{0}: \mu_{W} = \mu_{NW}$

$H_{a}: \mu_{W} \ne \mu_{NW}$

where $W$ is white mother and $NW$ is non-white mother. We can use the inference function again:
```
inference(y = nc$gained, x = nc$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 = 277, mean_not white = 28.6751, sd_not white = 15.1845
n_white = 694, mean_white = 30.9798, sd_white = 13.8234
```
```
Observed difference between means (not white-white) = -2.3047

H0: mu_not white - mu_white = 0 
HA: mu_not white - mu_white != 0 
Standard error = 1.052 
Test statistic: Z =  -2.19 
p-value =  0.0286 
```
If we assume 95% confidence level ($\alpha = 0.05$), we can see that the p-value is below the significance level. Therefore, we can reject the null in favor of the alternative, and state that there is sufficient evidence that the weight gained during pregnancy is different for white and non-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.

Inference for Numerical Data

Ryan Gordon

03/24/2019

North Carolina births

Exploratory analysis

Inference

On your own