For my research project I wanted to investigate the relationship between pet ownership and vegetarianism. I myself am vegetarian and own a pet which made this project appealing. In particular I wanted to conduct a study using the Islanders that would illustrate the relationship, if any, between owning a pet and being vegetarian. The population parameter that I wanted to study was the difference in proportions between pet owners who are vegetarian and pet owners who are not vegetarian. I relied on two sources in particular that motivated my research question; both were from the professional journal titled “Appetite: Multidisciplinary Research on Eating and Drinking.” The first journal article I read was written by Heiss and Hormes and addressed the relationship between pet ownership in childhood and likelihood of vegetarianism in adulthood. While my research question was not exactly the same, I addressed all ages and current pet ownership, this article suggested that there is indeed a relationship between childhood pet ownership and vegetarianism. The second piece of academic literature that I referenced was Hank Rothgerber’s piece regarding human diet, pet diet, pet ownership, and ethical dilemmas. In this study, the relationship between vegetarianism and owning a pet was studied. I decided to flip this research question into the one that I used for this research project, whether pet ownership made someone more likely to be vegetarian. Because of these two studies, I predicted that there would be a difference in proportions that was fairly large, around approximately 0.15. In this case, I was just interested in looking for a difference in proportions in any direction.

In order to conduct the study, I first needed to find a sample of Islanders that were willing to participate in my study. I found 471 Islanders from all three Islands that were willing to participate. I used random sampling and went through each city that was denoted by a circle and scrolled through and randomly selected a house and a person from within each house. I did not pay attention to age or any other variables, I just selected a random person from each house as I randomly scrolled. Each city was scrolled through and houses and people were randomly selected until all of the cities were sampled from and an adequate sample size was found. I sampled as many people as were possible in the time that I devoted to sampling, and the only thing that seemed to go wrong was participates dying as I did not pay attention to age or health. Due to the deaths of a few consenting participants, this brought my number of participants down to 471 from around 480.

Before I used R to perform any calculations, I decided what my observational units and variables were. My observational units are each person who participated in this study, as they are the ones who answer each question. For the variables,I chose to use two binary categorical variables; whether or not you own a pet and if you are vegetarian. The explanatory variable is whether or not someone owns a pet and the response variable is whether or not you are vegetarian. The results of the responses from the Islanders can be summarized in two different Two-Way tables, one representing counts, and one showing proportions.

Entering my data into R:

Vegetarian <- rbind(
do(47) * data.frame(diet = "Is Vegetarian", pet = "Has a Pet"),
do(31) * data.frame(diet = "Is Vegetarian", pet = "Does Not have a Pet"),
do(227) * data.frame(diet = "Is not vegetarian", pet = "Has a Pet"),
do(166) * data.frame(diet = "Is not vegetarian", pet = "Does Not have a Pet")
)

Vegetarian<-Vegetarian %>% 
  mutate(proportion=.row/ n()) %>% # creating a new variable called proportion
  select(pet,diet,proportion)

head(Vegetarian,n=3)

Finding the difference in proportions:

#sample sizes
n.pet<- 274 
n.nopet<- 197  

# counts
pet.vegetarian<-47
nopet.vegetarian<- 31

#sample proportions 
p.hat.pet <- pet.vegetarian/n.pet 
p.hat.nopet <- nopet.vegetarian/n.nopet  

# difference between the sample proportions of not having a pet vs having a pet
p.hat.diff<-p.hat.pet-p.hat.nopet
cat("difference in sample proportions of people who are vegetarian between non pet owners and pet owners is",p.hat.diff)
## difference in sample proportions of people who are vegetarian between non pet owners and pet owners is 0.01417244

This table shows my results in a two-way table with counts:

tally(diet ~ pet, data = Vegetarian, format = "count", margin = TRUE)
##                    pet
## diet                Does Not have a Pet Has a Pet
##   Is not vegetarian                 166       227
##   Is Vegetarian                      31        47
##   Total                             197       274

This table shows my results in a two-way table with proportions:

tally(diet ~ pet, data = Vegetarian, format = "prop")
##                    pet
## diet                Does Not have a Pet Has a Pet
##   Is not vegetarian           0.8426396 0.8284672
##   Is Vegetarian               0.1573604 0.1715328

In order to best represent my data graphically, I chose to use a stacked bar plot and a side by side bar plot. I feel that these best show the proportions of interest in this study and are easy to visually follow. Both of these plots illustrate that there does not appear to be any significant difference in proportions just by looking at the tables and graphs.

Stacked bar plot:

bargraph(~pet, groups = diet, data = Vegetarian, auto.key = TRUE, stack = TRUE)

Side by Side bar plot:

bargraph(~pet, groups = diet, data = Vegetarian, auto.key = TRUE)

The initial results of the study do not appear to show any significant difference in proportions between pet ownership and vegetarianism. At this point, it appears equally likely to own a pet and be vegetarian as it is not to own a pet and be vegetarian. Once again, my population parameter in this study is the difference in proportions between people who own a pet and are vegetarian and people who do not own a pet and are vegetarian. Going into my analysis of these results and with the population parameter in mind, my hypotheses were as follows:

Null Hypothesis: There is no significant difference between the proportion of people who own pets and are vegetarian and people who do not own pets and are vegetarian.

\(H_0: \pi_{petowners}-\pi_{nonpetowners}=0\)

Alternative Hypothesis: There is a significant difference between the proportion of people who own pets and are vegetarian and people who do not own pets and are vegetarian

\(H_0: \pi_{petowners}-\pi_{nonpetowners}\neq0\)

In terms of errors that could have been made when conducting the study, there are two types of errors that could have been present, Type I and Type II. A type I error is thought of as when the null hypothesis is rejected, but the null hypothesis is actually true. A Type II error is when the null hypothesis is actually wrong, but is not rejected. In the instances of this study, a Type I error would mean that I decided that there was a significant difference in proportions between the people who own pets and are vegetarian and people who do not own pets and are vegetarian. However, this would be false, and there is no significant difference. A Type II error would be the opposite, I would fail to reject the null hypothesis, but the null hypothesis would actually be wrong. For example, there actually is a difference in proportions between people who own pets and are vegetarian and those who do not own pets and are vegetarian. Because the literature I read led be to believe that there would be a significant difference in proportions between the two groups, any error that was found in this study would most likely be a Type II error.

I believe that my results are representative of the population of interest because the observational units were found from every location indicated by a circle on the Islands, all ages, and from all over each of the circle cities. Because I used a very large sample size of all ages, I would be able to generalize that the results represent, on average, all ages of all of the circle cities within the Islands.

When analyzing my results, I had to decide whether to use a simulation based approach or a theory based approach for my analysis. In order to analyze two binary categorical variables, a two sample z-test would be the appropriate theory based approach to use. The validity conditions that needed to be met include having a large sample size or at least ten observations in each cell of the two way chart. My study met the validity conditions for a two sample z-test and so I proceeded with the theory based test.

The first step I took was to find the standardized statistic using R in the code shown below:

#pooled proportion (total successes/combined sample size)
p.hat.pooled<- (pet.vegetarian+nopet.vegetarian)/(n.pet + n.nopet)

# Standard error of the theoretical sampling distribution based on a pooled proportion
SE.diff.null<-sqrt((p.hat.pooled*(1-p.hat.pooled))/n.pet 
                   + (p.hat.pooled*(1-p.hat.pooled))/n.nopet)

# Standardized statistic, z
z <- p.hat.diff/SE.diff.null 
cat("standardized statistic z is",round(z,2))
## standardized statistic z is 0.41

The standardized statistic z is equal to 0.41, which means that the difference in proportions between people who own pets and are vegetarian and people who do not own pets are are vegetarian is 0.41 standard deviations away from the null hypothesis.

Next, I found the two sided p-value using R in order to find the probability of observing a value at least as extreme as the difference in proportions (0.014) assuming that the null hypothesis is true.

#Theory-based test p-value
#left.tailed.p.value<-pnorm(z, 0, 1, lower.tail = TRUE) 
right.tailed.p.value<-pnorm(z, 0, 1, lower.tail = FALSE) 
#right.tailed.p.value

#equivalently:
right.tailed.p.value2<-pnorm(p.hat.diff, 0, SE.diff.null, lower.tail = FALSE)
#right.tailed.p.value2

two.sided.p.value<-2*right.tailed.p.value
cat("two-sided p-value is",two.sided.p.value)
## two-sided p-value is 0.6831635

The two-sided p-value that was found is 0.6831635 which is very far away from the typical cut off of significance for a p-value of 0.05. As of right now, by looking at both the z-statistic and the p-value, it can be seen that there does not appear to be a significant difference in proportions. This leads me to fail to reject the null hypothesis because neither the p-value nor the z-statistic provide enough evidence against the null hypothesis. In conclusion, there is no significant evidence that shows a significant difference between the proportion of people who own pets and are vegetarian, and people who do not own pets and are vegetarian.

Because we have failed to reject the null hypothesis, the next step to take will be to consider the range of values that the population parameter could fall into. I will use a 95% confidence interval to find these values.

# Standard error of the sampling distribution based on individual proportions

SE.diff.CI<-sqrt((p.hat.pet*(1-p.hat.pet))/n.pet
              + (p.hat.nopet*(1-p.hat.nopet))/n.nopet)


# margin of error for 95% CI
MoE <- 1.96 * SE.diff.CI
#MoE

LB<-p.hat.diff - MoE # lower limit of 95% CI
UB<-p.hat.diff + MoE # upper limit of 95% CI
round(cbind(LB,UB),2)
##         LB   UB
## [1,] -0.05 0.08

The 95% confidence interval is from -0.05 to 0.08 which means that we are 95% confident that the difference in population proportions between pet owners who are vegetarian and non-pet owners who are vegetarian are between -0.05 and 0.08. The initial difference in population proportions that was found is 0.014, which fits well within this range of values. In addition, the confidence interval contains zero. This is important because if the confidence interval contains zero, this is the value of the null hypothesis (\(H_0: \pi_{petowners}-\pi_{nonpetowners}=0\) ). Because the value of the null hypothesis is included in the confidence interval, this leads us to further conclude that we fail to reject the null hypothesis. I continue to fail to reject the null hypothesis as I found earlier when analyzing the results of the p-value and z-statistic.

In conclusion, I fail to reject the null hypothesis and there is no significant difference in proportions between people who are pet owners and vegetarian and people who are not pet owners and vegetarian. The two sided p-value of 0.68, z-statistic of 0.14, and confidence interval from -0.05 to 0.08 all lead me to draw this conclusion. The results of this study can be generalized to the population from which it was sampled because of the large sample size and random sampling that was used. This was an unexpected result after reading the two difference pieces from the “Appetite” journal who spoke quite differently. If I could conduct this study again, I would want to localize it to the citizens from one particular Island to account for potential differences in culture, health, or economic situation. Vegetarianism is something that differs from household to household, as well as within the income bracket that allows for a more plant based diet. These are factors I would like to consider if I was conducting this study again. I would also like to include more information on the type of pet that was owner and potentially address whether having a cat/dog versus a reptile could inform the diet someone chooses.

Bibliography:

Heiss, Sydney, and Julia M Hormes. “Ethical concerns regarding animal use mediate the relationship between variety of pets owned in childhood and vegetarianism in adulthood.” Appetite: Multidisciplinary Research on Eating and Drinking, vol. 123, 1 Apr. 2018, pp. 43–48, https://doi.org/https://doi.org/10.1016/j.appet.2017.12.005.

Rothgerber, Hank. “A meaty matter. Pet Diet and the vegetarian’s dilemma.” Appetite: Multidisciplinary Research on Eating and Drinking, vol. 68, 1 Sept. 2013, pp. 76–82, https://doi.org/https://doi.org/10.1016/j.appet.2013.04.012.