What statistical test to use to assess whether a medical procedure is related to recurrent disease? I have a question as to which statistical test I need to use to accomplish this.
I have a population of 80 patients:


*

*37 had procedure A 

*43 had procedure B


Of the 80 patients, 19 patients had recurrent disease:


*

*13 had procedure A

*6 had procedure B


How do I determine if having procedure A increases the risk of having
recurrent disease is statistically significant, given the differences
in numbers for procedure A and B?
 A: A contingency table analysis will suffice.
As an example, in Stata,
 tabi 13 24 \ 6 37, exact chi2

           |          col
       row |         1          2 |     Total
-----------+----------------------+----------
         1 |        13         24 |        37 
         2 |         6         37 |        43 
-----------+----------------------+----------
     Total |        19         61 |        80 



  Pearson chi2(1) =   4.9272   Pr = 0.026
           Fisher's exact =                 0.036
   1-sided Fisher's exact =                 0.025

The two groups are likely to really be different, and this difference is statistically significant.
Fisher's exact test P value = 0.036
Chi-squared P value = 0.026
95% confidence intervals of the difference can also be obtained easily:
csi 13 6 24 37, exact

                 |   Exposed   Unexposed  |      Total
-----------------+------------------------+------------
           Cases |        13           6  |         19
        Noncases |        24          37  |         61
-----------------+------------------------+------------
           Total |        37          43  |         80
                 |                        |
            Risk |  .3513514    .1395349  |      .2375
                 |                        |
                 |      Point estimate    |    [95% Conf. Interval]
                 |------------------------+------------------------
 Risk difference |         .2118165       |    .0263769     .397256 
      Risk ratio |         2.518018       |    1.063687    5.960791 
 Attr. frac. ex. |         .6028623       |    .0598736     .832237 
 Attr. frac. pop |         .4124847       |
                 +-------------------------------------------------
                                  1-sided Fisher's exact P = 0.0250
                                  2-sided Fisher's exact P = 0.0356

A: I figured I'd add the R version of @propofol's solution since it's free and not everyone has access to Stata:
# Prepare 2x2 table
total = c(37, 43)
recurrent = c(13, 6)
no_recurrence = total - recurrent

# Create the table
outcome_table = cbind(recurrent, no_recurrence)

# Add names to enhance understanding
rownames(outcome_table) <- c("Group A", "Group B")

# Show the table and see that it's correct
outcome_table

# Do the Pearson's Chi-square test with Yates' correction
chisq.test(outcome_table)

# Since 6 is almost 5 Fisher's exact test might be more suitable
fisher.test(outcome_table)

# Load library for a more fancy version
library(gmodels)

# Reminds more of the stata output
CrossTable(outcome_table, fisher=TRUE, chisq=TRUE)

A: While @propofol's answer is correct for your given scenario with no covariates, if you wish to extend your analysis to include covariates like patient characteristics, contingency table analysis quickly becomes cumbersome.
From there, you'd probably use either binomial or logistic regression. Given the relatively high prevalence of recurrence in your data set (~24%), I would probably go with binomial regression, which will provide you with a better estimate. But really, if you want to include covariates like that, my best advice is to go find a statistically-inclined epidemiologist or biostatistician.
