# Advice on the statistical analysis of clinical data using summary statistics, t-tests, ANOVA, and linear regression

I need some help with the statistical analysis of a study of a particular surgery to remove a particular cancer. I am using the statistical program R to conduct my analysis. My data are saved in the object study_data.

### Data

# Create reproducible example data
set.seed(50)

study_data <- data.frame(
Patient_ID = 1:500,
Institution = sample(c("New York","San Francisco","Houston","Chicago"),500,T),
Gender = sample(c("Male","Female"),500,T),
Race = sample(c("White","Black","Hispanic","Asian"),500,T),
Pathologic_stage = sample(c("P0","Pa","Pis","P1","P2a","P2b","P3a","P3b","P4a","P4b"),500,T),
Treatment_arm = sample(c("One","Two","Three","Four"),500,T),
Surgery_age = round(runif(500,20,100)),
Nodes_removed = round(runif(500,1,130)))


Here is what the data look like:

# Peak at the first six lines of the data

Patient_ID   Institution Gender     Race Tumor_grade Pathologic_stage Treatment_arm Surgery_age Nodes_removed
1          1       Houston   Male Hispanic         One              P2b           Two          77           130
2          2 San Francisco Female Hispanic       Three               Pa           Two          38           112
3          3      New York Female    Black        Four               P0          Four          90            90
4          4       Chicago   Male Hispanic         Two              Pis          Four          46             4
5          5       Houston Female    Black        Four              P2a          Four          96           114
6          6      New York   Male    Black       Three              P3b          Four          92             7


### My interest

I am interested in learning more about what variables are associated with the number of lymph nodes removed during the surgery. My first thought was to simply stratify the data by a particular variable and then calculate the median number of nodes removed.

For example, to see if the institution at which the surgery was performed mattered, I could write:

cbind(do.call(rbind, by(study_data$Nodes_removed, study_data$Institution, summary)))

Min. 1st Qu. Median  Mean 3rd Qu. Max.
Chicago          1   25.50   65.5 64.48   98.75  129
Houston          1   40.00   71.0 69.26  100.00  130
New York         4   36.00   67.0 67.96  100.00  129
San Francisco    3   36.75   61.0 65.76   99.00  127


This lets me compare the median nodes removed in each institutional city.

### My question

I would like to fully examine the association between all of my variables and the outcome Nodes_removed.

1. Should I just do these simple summary statistics for all of my variables?
2. Do I need to perform some sort of hypothesis test for all of the associations to say whether or not the summary statistics differ? For example, should I calculate a median and a confidence interval for each comparison?
3. Or should I be using t-tests to compare one group to another?
4. In the case of a multi-level variable, should I use ANOVA?
5. Is there any role for linear regression analysis here?
6. If I wanted to build a single model that includes every possible predictor variable, what method should I use?

For example, say that I am most interested in the association between the age at which the surgery was performed, Surgery_age, and Nodes_removed. However, I would like to adjust this association for potential confounders like gender, race, tumor grade, treatment arm, etc. What is the best way for me to do this?

Thanks for any advice you can give!

• My initial reaction was to start with a glm(..., family=poisson), b/c you have count data. I would start with summary(glm(Nodes_removed~(.), data=study_data, family=poisson)), but I notice the QQ plot doesn't look so great (if you plot the glm() object you get diagnostics). Also, there are a lot of terms in there, and I don't know what types of interactions etc. seem sensible. With data like this, I would do careful thinking about what makes sense, b/c blindly including a lot of terms can be dangerous. Commented May 7, 2013 at 0:52

1. Summary statistics are always useful, it's good to try to understand your data rather than rely merely on tests and p-values. Do report them, look at the data, use graphs and think hard about what it all means.

2. What you should do depends on your objectives (decision to go on with the treatment? publication? student paper?) Broadly speaking, the main reason to perform statistical inference in clinical trials is that such data are very noisy. Therefore, you would expect some differences to occur by chance and you want ways to sort out what could be the result of sampling variability, how big the effect of your treatment could be and whether the effect you see is likely to generalize well beyond your study. Without that, you run the risk of over-interpreting the pattern of results in your particular sample.

3. All the techniques you mention are in fact closely related but they are problematic for count data. Also, considering each variable individually can be misleading (especially those that obviously cannot be randomized like age, gender, race…). For example, it's entirely possible to have a large and significant difference between race as revealed by a one-way ANOVA that disappears once you consider, say, pathologic stage (e.g. because some group seeks treatment later). Detecting and interpreting this kind of things is not trivial.

4. See 3

5. See 3

6. This looks like the best approach and as explained by @rbatt, Poisson regression/Generalized linear models would be a good guess but building and understanding such models is a large and complex area. Knowing what to look for should give you a good starting point to find books on the topic or ask/read more specific questions on this site but you should not expect to go from wondering if a t-test can be used to compare groups to competently analyze a complex clinical trial in a few hours. If your main objective is getting results ASAP rather than learning, seeking advice from a more experienced researcher, or better yet, a statistician (either by hiring a consultant or checking if your employer already has statisticians on staff) is probably the only reasonable solution.

• Thanks very much for your thoughtful answer. I am going to have a hand at implementing the suggestions the two of you have made, and may post a new, more focused question in the future. Sorry for my delayed response, as well - things have been hectic this week. Commented May 8, 2013 at 18:38