Relationship Between Gun Ownership and Ethnicity Groups

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Contingency Table (Proportions)

prop.table(mytable, 2)

##

## White Non-Hispanic Black Non-Hispanic Hispanic Other Non-Hispanic

## Yes 0.4037212 0.1661601 0.1828396 0.2954545

## No 0.5962788 0.8338399 0.8171604 0.7045455

A graphical representation could be very useful in order to fully understand the scope of these data.

mosaicplot(prop.table(mytable, 2),las=2, main="Mosaic Plot Gun Ownership by Ethnicity Group")

Both the tables and the mosaicplot show that the conditional probabilities vary sensibly depending on race and ethnicity group.

The probability of gun ownership in the household given that the sampled respondent is white non-Hispanic is equal to roughly 40.3%.

The probability of gun ownership in the household given that the sampled respondent is black non-Hispanic is equal to roughly 16.7%.

The probability of gun ownership in the household given that the sampled respondent is Hispanic is equal to roughly 18.3%.

The probability of gun ownership in the household given that the sampled respondent belongs to a different ethnicity group from the ones listed is roughly 29.5%.

This suggests that gun ownership in the household and race or ethnicity are most likely dependent. At this stage of the analysis we are still working with the sample data therefore we can not extend the findings of the study to the population at large. However, the large differences between the conditional probabilities observed in the sample can be interpreted as a strong evidence that there is actually a real difference in the population of interest.

Inference:

The ultimate goal of our study is to establish if there is a real relationship between gun ownership in the households of US citizens aged 18 or older and ethnicity groups. We’re going to use the data we collected to verify if the gun ownership rate varies by ethnicity groups. In order to do so we have to perform a hypothesis test.

HYPOTHESIS TEST (FRAMEWORK)

In a hypothesis test we set 2 hypotheses: the Null Hypothesis and the Alternative Hypothesis. In the context of this study the hypotheses are:

Null Hypothesis: gun ownership in the households of US citizens aged 18 or older and ethnicity groups are independent of each other. Gun ownership rate does not vary by ethnicity group.

Alternative Hypothesis: gun ownership in the households of US citizens aged 18 or older and ethnicity groups are dependent of each other. Gun ownership rate varies by ethnicity group.

We want to test if gun ownership in the households of US citizens aged 18 or older and ethnicity groups are associated at the 5% significance level.

CHI-SQUARE INDEPENDENCE TEST (CONDITIONS)

Since we are evaluating the relationship between 2 categorical variables and one of which has more than 2 levels we are going to use the Chi-Square Independence Test. This test provides significant results if specific conditions are met. For this reason we must first check for them. The conditions are:

Indipendence: ANES data consists in a random sample from less than 10% of the population; each case only contributes to one cell in the table. We can assume that the sampled observations are independent.

Sample Size: since the overall gun ownership rate is 31.8% and the sample size is large, each particular scenario has more than 5 expected cases.

CHI-SQUARE INDEPENDENCE TEST (TEST STATISTIC & P-VALUE)

The Chi-Square Independence Test evaluates the hypotheses by quantifying how different the observed counts are from the expected counts. Large Deviations from what we would expect provide strong evidence for the Alternative Hypothesis. Calculations for the Chi-Square test statistic are tedious and error-prone, for this reason we are going to make use of computation.

chisq.test(mytable)

##

## Pearson’s Chi-squared test

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## data: mytable

## X-squared = 302.9267, df = 3, p-value < 2.2e-16

We can already anticipate something about the P-Value. We obtained a particularly high test statistic (X-squared = 302.9267), consequently the P-Value is bound to be very small. Another time computation comes to our rescue, we only need to plug in the test statistic (X-squared = 302.9267) and the degrees of dreedom (df = 3).

pchisq (302.9267, 3, lower.tail = FALSE)

## [1] 2.313867e-65

P-Value is almost 0 and is smaller than the significance level (0.05). Therefore we reject the null Hypothesis and conclude that the data do provide convincing evidence that there is a relationship between gun ownership in the households of US citizens aged 18 or older and ethnicity groups.

Conclusion:

It is useful to remember that this is an observational study therefore the type of the analysis is not sufficient to deduce a causal relationship. What is more we need to consider the effect of possible confounders; for instance the gun ownership could also be affected by the income level. We also noticed in the data exploration that the gun ownership rate was much higher for the respondents belonging to the “White Non-Hispanic Group”. Trying to understand the reason why could be the subject of possible research in the future.

References:

DATA REFERENCES

American National Elections Study, 2012 Time Series Study Coursera extract modified for Data Analysis and Statistical Inference course (Duke University).

R dataset can be downloaded