1) Introduction

In this study, I will studying the relation between the temperature at where a cancer patient is treated and the survival time (in days) of the patient.

Research question: ‘Does the patient living in warmer area survive cancer better ?’.

Parameter of interest: We are interested in the long-run average survival time after cancer diagnosis of patients.

Population: Cancer patients on the islands.

Mini Literature Review: I want to study a renowned relation between cancer mortality rate (survival time after cancer diagnosis) and the weather. In the research paper A link between cold environment and cancer, the author determined that patients in warmer area have a higher long-run average survival time after cancer diagnosis, this inspired me to do the study again on the simulation ‘The Island’.

Prediction: Before collecting the data, I believe that I will get the same result as the authors in the paper A link between cold environment and cancer, that is I believe that I will find out that patients living in warmer areas have a higher long-run average survival time compares to cancer patients in colder areas.

2) Data collection

I will study patients on the two island Ironbard and Bonne Santé to compare their survival time. Using random sampling data collection and student t-distribution test to compare the mean between two group.

My observational units are long-term resident of Ironbard island and Bonne Santé island that diagnosed cancer and died from it.

Method:

The patient data was collected in the hall of most city in the two island. All the patient died and their survival time is recorded in the hall death records. We use death records from 350-374 for all patient to keep our research up to date. The cities that data are collected are: Ironbard: Hofn, Helvig, Blondous, Helluland, Bjurholm, and Vardo. Bonne Santé: Nidoma, Riroura, Gordes, Colmar, Eden, Vaiku, Kinsale, Valais, Pauma, Talu, and Mahuti. All city on Ironbard is included and 11/12 cities on Bonne Santé is included.

Eligibility to be included:

One reason I choose this eligibility criteria is that I get inspired from the paper ‘Brain cancer after radiation exposure from CT examinations of children and young adults: results from the EPI-CT cohort study’. In this paper, the authors study the chance of getting brain cancer after CT examinations and the authors remove the data that get cancer after first 5 years of CT exposure since it is possible that something wrong is causing the patients to get CT exams, it is just that the CT doesn’t find out the cause. So, I decided to remove the patients who don’t actually is a native to Bonne Santé to remove the cases that patient are diagnosed with cancer already and then come to Bonne Santé for treatment.

Data collection:

First, we set a goal to collect roughly 70 patient records on each island. Since there are 6 island on Ironbard, we need to collect 11 records on the island.

Second, we go to each city’s hall in the island and write down all the name of all the past patient and number it from 1 to n, where n is the number of patient who died because of cancer.

Third, we will use a random number generator to pick 11 patients from n patients and then those 12 patients go our data. If n < 12 then we collect all data of n patients, n < 12 in this case means there are less than 12 patients who died from cancer recorded in the city hall from year 350-374. We do similar to the 11 islands on Bonne Santé where we need to collect 6 patient record on each island.

Notes: The data we are collecting in the Google sheet are ‘Island’ (variable 1), ‘City’ (to keep track the city we have collected data during data collection process), ‘Name’ (patient name, keeping track to not write down a patient name twice, and for the random sampling with random number generator), and ‘Living time since first cancer diagnose’ (variable 2). We will omit the patients that doesn’t meet our eligibility criteria: that include living in the island less than 5 years before cancer diagnosis. We check the time the patient lived in the island by reviewing their summary.

Potential errors

A potential sampling error can occured during the process of writting down the name of all patient in a city hall and number it from 1 to n where the proccess is long and requires patience. However, I’m confident that I tried my best to do this step and errors are minimized.

A non-sampling error is that the data collection process exclude patients who are living at the time of data collection. This is a coverage error where a group is not included in the data collecting process. This non-sampling error might make it difficult to generalize this result to all the islanders.

Problem:

One problem I meet was some patient survived cancer and then they died of cancer many many years later. At first, it seems unreasonable for me to calculate their survival time as the time since they are first diagnosed with the first cancer because if they diagnosed and recover at 20 years old and then live until 100 years old then their survival time is 80 years old! And this might be bias because it depends on how long they live, not how good at they are surviving cancer. I thought of counting a recover as 140 days (5 years in the island) but then I think of this situation: an island with 18 cancer survivors and two people died immediately from cancer will be worse at surviving cancer compare to an island where 20 people live about 5 years after cancer. I think it is correct to accept outliers to preserve the integrity of the data.

3) Descriptive Statistics

The two variables:

My first variable is a binary categorical variable ‘Does the patient live in a cold weather area or a hot weather area ?’.This variable will be answer by another proxy variable, which is ‘Does the patient live on Ironbard island or Bonne Santé island ?’, because Ironbard is an island with cold weather and Bonne Santé is an island with hot weather.

My second variable is the survival time after diagnosed with cancer for each patient (recorded in days). The second variable is a quantitative variable.

Data:

Cancer_data <- read_csv("~/Stats with Apps/MATH247 MiniProject3 - Sheet1.csv")
Cancer_data

Visualization:

Seeing the favorite statistics:

favstats(Living_time ~ Island, data = Cancer_data)
histogram(~Living_time | Island, data = Cancer_data, main ='Histogram', width = 1, layout = c(1, 2))

bwplot(Island ~ Living_time, horizontal = TRUE, main = 'Side by side boxplot', data = Cancer_data)

Ironbard statistics

\(n_{Ironbard}\)= 66

\(\bar{x}_{Ironbard}\)= 151.55

\(SD_{Ironbard}\)= 338.55

Bonne Santé statistics

\(n_{Bonne Santé}\)= 67

\(\bar{x}_{Bonne Santé}\)= 149.65

\(SD_{Bonne Santé}\)= 293.8

Based on the data, we see that \(\bar{x}_{Ironbard} = 151.55\) and \(\bar{x}_{Bonne Santé} = 149.65\). With that in mind, we have \(\bar{x}_{Ironbard} - \bar{x}_{Bonne Santé}= 1.9\).

Some comments:

Based on the histogram, the side-by-side boxplot, and the we see that there are quite a lot of outliers in the data, and there are outliers in both groups.

Based on the boxplots, we see that the Ironbard island’s median is slightly higher than Bonne Sante’s island.

Based on the summary of statistics, we see that Bonne Santé’s island have a lower median. (\(\bar{x}_{Bonne Santé}\) < \(\bar{x}_{Ironbard}\)). Which is consistent with our side by side boxplots. However, the sample standard deviation of the Ironbard group is higher.

We see that \(\bar{x}_{Ironbard} > \bar{x}_{Bonne Santé}\) based on the statistics summary. Although it is only a little bit higher, the way we formulated our research question indicated that this statistics wouldn’t help us support our hypothesis in the research question. In context, our research question suspected that there is a higher long-run average survival time in cancer patients on Bonne Santé island, but we have a sample indicate that the long-run average survival time in cancer patients on Ironbard island is higher.

4) Analysis of results

Population: Population is all the native cancer patients on Ironbard and Bonne Santé island.

This population definition is different from the population definition in the beginning that it included the name of specific island and the word ‘native’. This is because based on our data collection, we specifically choose two islands that have a distinguishable warm weather and cold weather. I included ‘native’ because they also must have lived on the island more than 5 years before diagnosed with cancer to be included in the study.

Parameters of interest: the long-run average survival time after cancer diagnosis

Null and alternative hypothesis:

Our hypothesis are:

Null hypothesis: There is no different between the long-run average survival time between cancer patients on Ironbard island and Bonne Santé island.

\[H_0: \mu_{Ironbard} - \mu_{Bonne Santé} = 0\]

Alternative hypothesis: The long-run average survival time between cancer patients on Ironbard island is lower than than the long-run average survival time between cancer patients on Bonne Santé island. \[H_a:\mu_{Ironbard} - \mu_{Bonne Santé} < 0\]

Errors possibilities:

Type I error: rejecting the null hypothesis when the null hypothesis is true. This happens when there is actually no difference in the long-run average survival time between cancer patients on Ironbard and Bonne Santé island, but we found that the long-run average survival time between cancer patients on Ironbard island is lower than long-run average survival time between cancer patients on Bonne Santé island. We can minimize the chance of getting type I error by decreasing confidence level.

Type II error: not rejecting the null hypothesis when the null hypothesis is false. This happens when the long-run average survival time of cancer patients on Bonne Santé island is actually higher than the long-run average survival time of cancer patients on Ironbard island but we don’t reject the null hypothesis, meaning we don’t find enough evidence that there is a difference in the long-run average survival time between cancer patients on Ironbard and Bonne Santé island.

In our case, I’m more worried about Type II error since our sample size is not very big.

Discussion about measurements

My measurement used random sampling in the data collection process and also incorporate data from nearly all cities in the two island so I think my measurement is representing the population I stated at the beginning of part 4). I would argue that my measurement can be generalized to all native islanders of Ironbard and Bonne Santé, who have lived on their island for a long time and have the same metabolism as other people on their islands.

One weakness in my data collection is that the sample size is not very big.

Another weakness is that my data doesn’t record those who are living with cancer. However, I would argue that my study’s parameter of interest is the long-run average survival time after cancer diagnosis, so currently those who are alive we don’t know what is there survival time after diagnosis because there survival time is still increasing. So those who are surviving seems not to carry the parameter of interest that my study is looking for.

Theory-based approach

We will first find p-value using theory-based approach. I first want to find out if the null distribution generated by the t.test function in R is taking the difference mean of \(\mu_{Ironbard} - \mu_{Bonne Santé}\) or \(\mu_{Bonne Santé} - \mu_{Ironbard}\) as the center for the null hypothesis so I will do a small check.

t.test(Living_time ~ Island, data = Cancer_data)
## 
##  Welch Two Sample t-test
## 
## data:  Living_time by Island
## t = -0.034595, df = 128.96, p-value = 0.9725
## alternative hypothesis: true difference in means between group Bonne Santé and group Ironbard is not equal to 0
## 95 percent confidence interval:
##  -110.6043  106.8029
## sample estimates:
## mean in group Bonne Santé    mean in group Ironbard 
##                  149.6515                  151.5522

Here, we see that the confidence interval is [-110.6,106.8], and |-110.6| \(>\) 106.8, meaning the confidence interval takes the value of \(\bar{x}_{Bonne Santé} - \bar{x}_{Ironbard}\) as the center. This indicates that the center of the null distribution is \(\mu_{Bonne Santé} - \mu_{Ironbard}\). We are interested in \(\mu_{Ironbard} - \mu_{Bonne Santé} < 0\) which is \(\mu_{Bonne Santé} - \mu_{Ironbard} > 0\). Therefore, we will conduct a right tail test. And we will do that by adding alternative = “greater” to our t.test function.

stat(t.test(Living_time ~ Island, data = Cancer_data, alternative = "greater"))
##           t 
## -0.03459536
pval(t.test(Living_time ~ Island, data = Cancer_data, alternative = "greater"))
##   p.value 
## 0.5137721

Our t-statistics is -0.034595.

Our p-value is 0.5138.

Intepretation: assuming the null hypothesis is true, that is there is no difference between long-run average cancer survival time after diagnosis between cancer patients on Ironbard and Bonne Santé island, there is a 51.38% we observe the statistics we collected (which is quite probable).

So assuming that there is no difference in average mean of cancer survival time after cancer diagnosis between two group, there are 51.38% chance that we get the statistics that we got. This makes sense since our statistics difference is \(\bar{x}_{Bonne Santé} - \bar{x}_{Ironbard} = -1.9 < 0\) so we expect to get a p-value larger than 0.5 if we do a right-tailed test.

Conclusion: with a p-value of 0.5138, we don’t find evidence against the null hypothesis. We fail to reject \(H_0\). Meaning that we don’t reject the hypothesis that the difference survival time after cancer diagnosis between people on Ironbard island and Bonne Santé island is 0.

Validity conditions for theory-based test

Simulation-based approach

First, we are going to do a difference mean shuffling to see if the null distribution generated by the diffmean function in R is taking the difference mean of \(\mu_{Ironbard} - \mu_{Bonne Santé}\) or \(\mu_{Bonne Santé} - \mu_{Ironbard}\) as the center for the null hypothesis.

set.rseed(498) 
Cancer.null <- diffmean(Living_time ~ Island, data = Cancer_data)
head(Cancer.null,2)
## diffmean 
## 1.900724

The difference mean is 1.9, clearly the difference mean is taking \(\bar{x}_{Ironbard} - \bar{x}_{Bonne Santé}= 1.9\) so we expect the simulation the null distribution of \(\bar{x}_{Ironbard} - \bar{x}_{Bonne Santé}\). So at the end, we will do a left-tail test since we are interested in \(\mu_{Ironbard} - \mu_{Bonne Santé} < 0\).

We generate a simulation:

set.rseed(498) 
Cancer.null <- do(1000)*diffmean(shuffle(Living_time) ~ Island, data = Cancer_data)
head(Cancer.null,2)

We view the distribution of the simulation when the null hypothesis is supposed to be true.

dotPlot(~ diffmean,
        data = Cancer.null,
        main="Simulated null distribution of the difference in sample means",
        xlab="difference in sample means",
        width = 2, cex = 1,
        groups = (diffmean >= 15.92))

p_value<-prop(~(diffmean <= 1.9), data = Cancer.null)
cat("left-tailed p-value is", p_value)
## left-tailed p-value is 0.506

Intepretation: assuming the null hypothesis is true, that is there is no difference between long-run average cancer survival time after diagnosis between cancer patients on Ironbard and Bonne Santé island, there is a 50.6% we observe the statistics we collected (which is quite probable).

Since the left-tailed p-value is 0.506, we don’t find evidence to reject the null hypothesis. We fail to reject \(H_0\). So we don’t find evidence to reject the hypothesis that long-run average survival time after cancer diagnosis is the same on Ironbard island and Bonne Santé island.

Confidence interval

Using the function t.test, our confidence interval is.

confint(t.test(Living_time ~ Island, data = Cancer_data))

The confidence interval is [-110.6,106.8]. We are 95% sure that the value of (long-run average survival time after cancer diagnosis in Bonne Santé) - (long-run average survival time after cancer diagnosis in Ironbard) is from -110.6 days to 106.8 days.

We are 95% sure that the long-run average survival time after cancer diagnosis in Bonne Santé is from 110.6 days shorter to 106.8 days longer compare to long-run average survival time after cancer diagnosis in Ironbard. Or we are are 95% sure that the long-run average survival time after cancer diagnosis on Ironbard is from 106.8 days shorter to 110.6 days longer compare to long-run average survival time after cancer diagnosis in Ironbard.

Since 0 is contained in the interval, it fits our conclusion to not reject the null hypothesis.

In the world of The Island, 106.8 days shorter is about 3.8 years shorter and 110.6 days longer is about 4 years longer so our confident interval is very wide.

Removing outliers data

We remove the outliers in our data to see if our hypothesis tests to see what happens to our confidence interval. We remove the data that have living days > 280 (10 years). This is because we want to remove the people who survived cancer from very young age and then died from cancer again.

Cancer_data_No_Outliers <- read_csv("~/Stats with Apps/MiniProject 3 _ NoOutliers - Sheet1.csv")
head(Cancer_data_No_Outliers,2)
histogram(~Living_time | Island, data = Cancer_data_No_Outliers,main = 'Histogram', width = 1, layout = c(1, 2))

bwplot(Island ~ Living_time, horizontal = TRUE, main = 'Side by side boxplot', data = Cancer_data_No_Outliers)

favstats(Living_time ~ Island, data = Cancer_data_No_Outliers)

Based on the histogram, the side by side box plots and the statistics summary, we see that there is much less variation in our data.

confint(t.test(Living_time ~ Island, data = Cancer_data_No_Outliers, conf.level = 0.95))

We can see that the confident interval now is much smaller. With the data without outliers, we are 95% confident that long-run average survival time on Bonne Santé island after cancer diagnosis is from -30.8 days shorter to 2.9 days longer compares to survival time on Ironbard island. However, 0 is still included in the interval so we can’t reject the hypothesis that there is no difference in long-run average survival rate between two groups is not rejected.

This interval is much smaller and can provide more useful information for those who are deciding a place to live.

5) Discussion

Behavior of data:

The data and the results didn’t behave as I expected as I expected to draw a conclusion similar to what the authors in the paper A link between cold environment and cancer did, which is I expected to conclude patients on Bonne Santé island have higher long-run average survival time.

Here are some reasons I think that will make my study’s result different:

Takeaways:

One lesson that I learnt is that sometime data are strongly skew. It is not a bad things but in this case it gives us a super large confidence interval with the original data. It is possible that we remove outliers to see what happen to the confidence interval but we can’t treat the original data like we didn’t collect them. It is necessary to report what we collected originally.

Generalization:

As I stated earlier, I think that my data is a representation of people living on Bonne Santé and Ironbard islands. I however feel that a negative aspect of my study is that my study uses a small sample. Further studies with larger cohort needed to evaluate the claim that colder weather associates with less cancer survival time.

A non-sampling error is also a potential stop to the generalization of my study. In the data collection process, I don’t include those who are cured from cancer and is alive. However, I already give an argument in the beginning of section 4) so I believe this is not a problem.

Anything I would do different

I’m curious of asking the question ‘Does warmer weather provides more chance to survive cancer ?’ and have my parameter of interests is the long-run proportion of cancer patients that survive cancer. If I compare the proportion of people surviving cancer (live more than 5 or 10 years) in two groups, perhaps I could find some interesting association. If I were do the study again, I would choose to ask this research question.

Similar questions

Some similar relations that someone might try to discover could be the homogeniety of average cancer survival time between cancer types in different weathers. For example, maybe in the warm weather it is better to survive some specific types of cancer, but the cold weather is better to survive some other specific types of cancer. Further studies needed to establish the relationship. The average survival time and average temperature per year is also an interesting relationship between two quantitative variables where average survival time is the response variable.

6) Bibliography

Sharma A, Verma HK, Joshi S, Panwar MS, Mandal CC, 2015. A link between cold environment and cancer. Tumour Biol.

Michael Hauptmann, ‘Brain cancer after radiation exposure from CT examinations of children and young adults: results from the EPI-CT cohort study’, THE LANCET Oncology, Volume 24, Pages 45-53, January 2023 (https://doi.org/10.1016/S1470-2045(22)00655-6)

RPub link: http://rpubs.com/phucphucphucphuc/1183169