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")
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?

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

summary(nc)
nrow(nc)  #this gives us the number of rows as well as the number of cases

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.

boxplot(nc$fage)

boxplot(nc$mage)

boxplot(nc$weeks)

boxplot(nc$visits)

boxplot(nc$gained)

boxplot(nc$weight)

#looking at the boxplots for the numerical categories, it looks as though they all have some outliers

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 boxplot shows that the median is about the same but non smokers have wider range.

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)

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 conditions for inference are independence/sample size and normality. Our sample size is from less than 10% of population. Because of our large sample size, we can relax the nearly normal condition.

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

NULL: ($\mu_{nonsmoker} - \mu_{smoker}$) = 0
ALTERNATE: ($\mu_{nonsmoker} - \mu_{smoker}$) != 0

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")

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", 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 )

#95% confidence interval is: ( 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"))

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(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 )

#95% CI = (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(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 )

#90% CI: (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" )

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

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

# our p-value, .8526 > .05 so we fail to reject the null hypothesis which was that there was no difference in the weights

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.

#create subsets of mature and younger moms and verify that this was done
mature <- subset(nc, mature == "mature mom") #
younger <- subset(nc, mature == "younger mom")
View(mature)
View(younger)
#find the max and min age for all the mature moms
maxmat.age <- max(mature$fage, na.rm = TRUE)
maxmat.age

## [1] 55

minmat.age <- min(mature$fage, na.rm = TRUE)
minmat.age

## [1] 26

#find the max and min age for all the younger moms
maxyng.age <- max(younger$fage, na.rm = TRUE)
maxyng.age

## [1] 48

minyng.age <- min(younger$fage, na.rm = TRUE)
minyng.age

## [1] 14

#This showed something quite interesting. I'm not certain what criteria was used to determine mature vs. younger but it does not seem to be  purely age since the range of both overlap.

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 like to explore if whether a baby is considered low birth weight is impacted by how much weight the mother gains during pregnancy. In plain terms, is there is a difference in average weight gained in mother with low birth weight babies and mothers with babies who are not considered low birth weight.
- NULL: ($\mu_{low} - \mu_{notlow}$) = 0
- ALT: ($\mu_{low} - \mu_{notlow}$) != 0

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

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_low = 104, mean_low = 26.0769, sd_low = 14.4065
## n_not low = 869, mean_not low = 30.8343, sd_not low = 14.1444

## Observed difference between means (low-not low) = -4.7574
## 
## H0: mu_low - mu_not low = 0 
## HA: mu_low - mu_not low != 0 
## Standard error = 1.492 
## Test statistic: Z =  -3.189 
## p-value =  0.0014

#our p-value, .0014 < .05 so we reject our null hypothesis meaning there is enough evidence in the data to say that there is some difference in the average weight gained by mothers whose babies were considered low birth weight and those not.

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.