Synthetic Data Exploratory Analysis

Load the packages.

# Set a seed for reproducibility
set.seed(234)
# Define the number of observations (patients) to generate
n_patients <- 100

# Create an empty data frame with placeholders for variables
syn_dat <- data.frame(
  PatientID = numeric(n_patients),
  Age = numeric(n_patients),
  Gender = character(n_patients),
  TreatmentGroup = character(n_patients),
  EnrollmentDate = lubridate::as_date(character(n_patients)),
  BloodPressure = numeric(n_patients),
  Troponin = numeric(n_patients),
  AdverseEvent = integer(n_patients)
)

# Variable 1: Patient ID
syn_dat$PatientID <- 1:n_patients

# Variable 2: Age (numeric variable)
syn_dat$Age <- round(rnorm(n_patients, mean = 60, sd = 10), 1)

# Variable 3: Gender (categorical variable)
syn_dat$Gender <- purrr::map_chr(sample(c("Male", "Female"), n_patients, replace = TRUE), as.character)

# Variable 4: Treatment Group (categorical variable)
syn_dat$TreatmentGroup <- purrr::map_chr(sample(c("A", "B", "Placebo"), n_patients, replace = TRUE), as.character)

# Variable 5: Date of Enrollment (date variable)
syn_dat$EnrollmentDate <- lubridate::as_date(sample(seq(from = lubridate::as_date("2022-01-01"), to = lubridate::as_date("2022-12-31"), by = "days"), n_patients, replace = TRUE))

# Variable 6: Blood Pressure (numeric variable)
syn_dat$BloodPressure <- round(runif(n_patients, min = 90, max = 160), 1)

# Variable 7: Troponin level (numeric variable)
# Option 1: Troponin level is independent of treatment
#syn_dat$Troponin <- round(rnorm(n_patients, mean = 0.10, sd = 0.15), 1)

# Option 2: Troponin is dependent on treatment
syn_dat$Troponin[syn_dat$TreatmentGroup == "A"] <- pmax(round(rnorm(sum(syn_dat$TreatmentGroup == "A"), mean = 0.035, sd = 0.012), 3), 0.001)
syn_dat$Troponin[syn_dat$TreatmentGroup == "B"] <- pmax(round(rnorm(sum(syn_dat$TreatmentGroup == "B"), mean = 0.085, sd = 0.020), 3), 0.001)
syn_dat$Troponin[syn_dat$TreatmentGroup == "Placebo"] <- pmax(round(rnorm(sum(syn_dat$TreatmentGroup == "Placebo"), mean = 0.260, sd = 0.060), 3), 0.001)

# Variable 8: Adverse Event (binary variable, 0 = No, 1 = Yes)
# Option 1: Adverse events are independent of treatment
#syn_dat$AdverseEvent <- purrr::map_int(sample(0:1, n_patients, replace = TRUE, prob = c(0.8, 0.2)), as.integer)

# Option 2: Adverse events are influenced by treatment status
syn_dat$AdverseEvent[syn_dat$TreatmentGroup == "A"] <- purrr::map_int(sample(0:1, sum(syn_dat$TreatmentGroup == "A"), replace = TRUE, prob = c(0.5, 0.5)), as.integer)
syn_dat$AdverseEvent[syn_dat$TreatmentGroup == "B"] <- purrr::map_int(sample(0:1, sum(syn_dat$TreatmentGroup == "B"), replace = TRUE, prob = c(0.7, 0.3)), as.integer)
syn_dat$AdverseEvent[syn_dat$TreatmentGroup == "Placebo"] <- purrr::map_int(sample(0:1, sum(syn_dat$TreatmentGroup == "Placebo"), replace = TRUE, prob = c(0.9, 0.1)), as.integer)

# Print the first few rows of the generated data
head(syn_dat)

  PatientID  Age Gender TreatmentGroup EnrollmentDate BloodPressure Troponin
1         1 66.6 Female              B     2022-04-22         121.6    0.063
2         2 39.5 Female        Placebo     2022-09-08         110.3    0.265
3         3 45.0 Female        Placebo     2022-05-15         100.4    0.277
4         4 74.7   Male        Placebo     2022-09-15         119.8    0.340
5         5 74.6   Male        Placebo     2022-06-27         127.7    0.191
6         6 61.4 Female              B     2022-11-14         121.6    0.093
  AdverseEvent
1            0
2            0
3            0
4            0
5            0
6            0

# Grab summary of synthetic data
summary(syn_dat)

   PatientID           Age              Gender      TreatmentGroup
 Min.   :  1.00   Min.   :29.60   Length   :100   Length   :100   
 1st Qu.: 25.75   1st Qu.:53.55   N.unique :  2   N.unique :  3   
 Median : 50.50   Median :61.70   N.blank  :  0   N.blank  :  0   
 Mean   : 50.50   Mean   :60.04   Min.nchar:  4   Min.nchar:  1   
 3rd Qu.: 75.25   3rd Qu.:66.15   Max.nchar:  6   Max.nchar:  7   
 Max.   :100.00   Max.   :81.10                                   
 EnrollmentDate       BloodPressure      Troponin       AdverseEvent 
 Min.   :2022-01-03   Min.   : 90.5   Min.   :0.0110   Min.   :0.00  
 1st Qu.:2022-04-25   1st Qu.:107.0   1st Qu.:0.0480   1st Qu.:0.00  
 Median :2022-07-15   Median :121.9   Median :0.0930   Median :0.00  
 Mean   :2022-07-06   Mean   :123.6   Mean   :0.1353   Mean   :0.31  
 3rd Qu.:2022-09-17   3rd Qu.:141.2   3rd Qu.:0.2320   3rd Qu.:1.00  
 Max.   :2022-12-27   Max.   :159.8   Max.   :0.4000   Max.   :1.00  

dplyr::glimpse(syn_dat)  

Rows: 100
Columns: 8
$ PatientID      <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …
$ Age            <dbl> 66.6, 39.5, 45.0, 74.7, 74.6, 61.4, 62.1, 29.6, 55.1, 4…
$ Gender         <chr> "Female", "Female", "Female", "Male", "Male", "Female",…
$ TreatmentGroup <chr> "B", "Placebo", "Placebo", "Placebo", "Placebo", "B", "…
$ EnrollmentDate <date> 2022-04-22, 2022-09-08, 2022-05-15, 2022-09-15, 2022-0…
$ BloodPressure  <dbl> 121.6, 110.3, 100.4, 119.8, 127.7, 121.6, 134.4, 107.4,…
$ Troponin       <dbl> 0.063, 0.265, 0.277, 0.340, 0.191, 0.093, 0.011, 0.235,…
$ AdverseEvent   <int> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0…

# Frequency table for adverse events stratified by treatment
table(syn_dat$AdverseEvent,syn_dat$TreatmentGroup)

   
     A  B Placebo
  0 11 26      32
  1 20  8       3

# Boxplot of Troponin levels by treatment group
ggplot(syn_dat, aes(x = TreatmentGroup, y = Troponin)) +
  geom_boxplot() +
  labs(title = "Troponin Levels by Treatment", x = "Treatment Group", y = "Troponin Level") +
  theme_bw()

# There is a difference in troponin levels across treatment groups, 
# with the placebo group having higher levels on average, and treatment group A has the lowest levels, which 
# correlates with lower risk of adverse events such as heart attacks.

# Perform ANOVA to test for differences in troponin levels across treatment groups
anova_result <- aov(Troponin ~ TreatmentGroup, data = syn_dat)
summary(anova_result)

               Df Sum Sq Mean Sq F value Pr(>F)    
TreatmentGroup  2 1.0088  0.5044   284.8 <2e-16 ***
Residuals      97 0.1718  0.0018                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# A significant p-value (p < 0.05) would indicate that there are significant differences in troponin levels 
# between at least two of the treatment groups. Our p-value is <2e-16, suggesting strong evidence for difference.

# We can also use linear models with treatment group as a predictor of troponin levels, and so forth.
# Model 1: treatment only
m1 <- lm(Troponin ~ TreatmentGroup, data = syn_dat)
summary(m1)

Call:
lm(formula = Troponin ~ TreatmentGroup, data = syn_dat)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.150914 -0.015161  0.000462  0.015059  0.131086 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           0.035161   0.007559   4.652 1.04e-05 ***
TreatmentGroupB       0.053780   0.010451   5.146 1.39e-06 ***
TreatmentGroupPlacebo 0.233753   0.010380  22.520  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.04208 on 97 degrees of freedom
Multiple R-squared:  0.8545,    Adjusted R-squared:  0.8515 
F-statistic: 284.8 on 2 and 97 DF,  p-value: < 2.2e-16

# Model 2: add demographics + blood pressure
m2 <- lm(Troponin ~ TreatmentGroup + Age + Gender + BloodPressure, data = syn_dat)
summary(m2)

Call:
lm(formula = Troponin ~ TreatmentGroup + Age + Gender + BloodPressure, 
    data = syn_dat)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.15481 -0.01436 -0.00023  0.01403  0.12574 

Coefficients:
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)            3.330e-02  3.888e-02   0.856    0.394    
TreatmentGroupB        5.376e-02  1.070e-02   5.023 2.42e-06 ***
TreatmentGroupPlacebo  2.338e-01  1.058e-02  22.108  < 2e-16 ***
Age                    1.627e-04  4.696e-04   0.347    0.730    
GenderMale             6.875e-03  8.566e-03   0.803    0.424    
BloodPressure         -9.185e-05  2.175e-04  -0.422    0.674    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.04254 on 94 degrees of freedom
Multiple R-squared:  0.8559,    Adjusted R-squared:  0.8483 
F-statistic: 111.7 on 5 and 94 DF,  p-value: < 2.2e-16

# Model 3: allow treatment effect to vary by blood pressure
m3 <- lm(Troponin ~ TreatmentGroup * BloodPressure + Age + Gender, data = syn_dat)
summary(m3)

Call:
lm(formula = Troponin ~ TreatmentGroup * BloodPressure + Age + 
    Gender, data = syn_dat)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.155029 -0.014499  0.001073  0.014565  0.121596 

Coefficients:
                                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)                          3.947e-03  5.959e-02   0.066    0.947    
TreatmentGroupB                      5.510e-02  6.584e-02   0.837    0.405    
TreatmentGroupPlacebo                3.139e-01  7.137e-02   4.398 2.93e-05 ***
BloodPressure                        1.076e-04  3.966e-04   0.271    0.787    
Age                                  2.547e-04  4.768e-04   0.534    0.594    
GenderMale                           5.505e-03  8.688e-03   0.634    0.528    
TreatmentGroupB:BloodPressure       -8.076e-06  5.308e-04  -0.015    0.988    
TreatmentGroupPlacebo:BloodPressure -6.394e-04  5.652e-04  -1.131    0.261    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.0426 on 92 degrees of freedom
Multiple R-squared:  0.8586,    Adjusted R-squared:  0.8479 
F-statistic: 79.81 on 7 and 92 DF,  p-value: < 2.2e-16

# Compare fit
anova(m1, m2, m3)

Analysis of Variance Table

Model 1: Troponin ~ TreatmentGroup
Model 2: Troponin ~ TreatmentGroup + Age + Gender + BloodPressure
Model 3: Troponin ~ TreatmentGroup * BloodPressure + Age + Gender
  Res.Df     RSS Df Sum of Sq      F Pr(>F)
1     97 0.17180                           
2     94 0.17009  3 0.0017082 0.3138 0.8153
3     92 0.16692  2 0.0031708 0.8738 0.4208

AIC(m1, m2, m3)

   df       AIC
m1  4 -344.8713
m2  7 -339.8706
m3  9 -337.7523

Across models, Treatment Group was the main predictor of Troponin levels, with Age and Blood Pressure also contributing in more complex models. Group B tended to trend higher, with 0.054 ng/mL higher troponin, and Placebo was 0.234 ng/mL higher comparatively. This means that those taking the placebo have a higher risk of heart attack or injury.

Note: This is a simulated dataset for educational purposes only and does not represent real patient data. Chatgpt-5 mini was used to help generate this code within Positron. All text descriptions are my own.