Summary the Data

Mean age, prostate volume, and preoperative PSA, broken down by race and family history of prostate cancer.

prostate_factors %>%
  select(age, p_vol, preop_psa, aa, fam_hx) %>%
  group_by(aa, fam_hx) %>%
  summarize(across(age:preop_psa, ~ mean(.x, na.rm = TRUE)), .groups = "drop") %>%
  kable(digits = 2, caption = "Mean values by race and family history") %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Mean values by race and family history
aa fam_hx age p_vol preop_psa
White No Family History 61.82 56.89 8.06
White FHx of Prostate Cancer 59.48 57.29 7.22
African American No Family History 60.70 54.35 9.90
African American FHx of Prostate Cancer 60.09 51.41 8.71

Visualization the Data

Scatterplot of preoperative PSA against prostate volume, with linear fits, faceted by race (rows) and family history (columns).

ggplot(prostate_factors) +
  aes(x = p_vol, y = preop_psa, col = aa) +
  geom_point() +
  geom_smooth(method = "lm") +
  facet_grid(aa ~ fam_hx) +
  labs(x = "Prostate Volume", y = "Preoperative PSA",
       title = "Relationship between Prostate Volume and Preoperative PSA,\nSubdivided by Family History and Race") +
  theme(legend.position = "bottom")

Statistical Testing of Differences

We hypothesize that mean preoperative PSA differs between African-American and White patients. H0: no difference. H1: a difference exists. We test this with an independent-samples (Welch’s) Student’s t-test.

prostate_factors %>%
  t_test(formula = preop_psa ~ aa, detailed = TRUE) %>%
  kable(digits = 2, caption = "T-test: preoperative PSA by race") %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
T-test: preoperative PSA by race
estimate estimate1 estimate2 .y. group1 group2 n1 n2 statistic p df conf.low conf.high method alternative
-1.89 7.86 9.75 preop_psa White African American 259 54 -1.96 0.05 71.69 -3.81 0.03 T-test two.sided

African-American patients had a higher observed mean preoperative PSA (9.75) than White patients (7.86), a difference of 1.89 units. However, this difference did not reach statistical significance (p = 0.053), and the 95% confidence interval for the difference (−3.81 to 0.03) narrowly crosses zero. The African-American subgroup was small (n = 54) relative to the White group (n = 259), so this analysis is likely underpowered. A larger, more balanced sample would be needed to determine whether this difference is real.