# Targeted logistic model coefficients

In `simstudy`, there are at least two ways to define a binary data generating process. The first is to operate on the scale of the proportion or probability using the identity link. This allows users to define a data generating process that reflects assumptions about risk ratios and risk differences when comparing two groups defined by an exposure or treatment. However, this process can become challenging when introducing other covariates, because it can be difficult to constrain the probabilities so that they fall between 0 and 1.

The second approach works on the log-odds scale using a logit link, and is much more amenable to accommodating covariates. Unfortunately, this comes at the price of being able to easily generate specific risk ratios and risk differences, because all parameters are log-odds ratios. The overall (marginal) prevalence of an outcome in a population will vary depending on the distribution of covariates in that population, and the strengths (both absolute and relative) of the association of those covariates with the outcome. That is, the coefficients of a logistic model (including the intercept) determine the prevalence. The same is true regarding the risk ratio and risk difference (if there is one particular exposure or treatment of interest) and the AUC.

Here we start with the simplest case where we have a target marginal proportion or prevalence, and then illustrate data generation with three other target statistics: risk ratios, risk differences, and AUCs.

### Prevalence

In this first example, we start with one set of assumptions for four covariates \(x_1, x2 \sim N(0, 1)\), \(b_1 \sim Bin(0.3)\), and \(b_2 \sim Bin(0.7)\), and generate the outcome y with the following data generating process:

\[ \text{logit}(y) = 0.15x_1 + 0.25x_2 + 0.10b_1 + 0.30b_2\]

    library(simstudy)
    library(ggplot2)
    library(data.table)

    coefs1 <- c(0.15, 0.25, 0.10, 0.30)

    d1 <- defData(varname = "x1", formula = 0, variance = 1)
    d1 <- defData(d1, varname = "x2", formula = 0, variance = 1)
    d1 <- defData(d1, varname = "b1", formula = 0.3, dist = "binary")
    d1 <- defData(d1, varname = "b2", formula = 0.7, dist = "binary")

    d1a <- defData(d1, varname = "y",
      formula = "t(..coefs1) %*% c(x1, x2, b1, b2)",
      dist = "binary", link = "logit")

    set.seed(48392)

    dd <- genData(500000, d1a)
    dd

    ##             id    x1     x2 b1 b2 y
    ##      1:      1  0.29  0.390  0  1 1
    ##      2:      2  0.76 -0.925  0  0 0
    ##      3:      3 -1.47  0.939  0  0 1
    ##      4:      4  1.92  0.560  0  1 1
    ##      5:      5  1.40 -0.238  0  1 0
    ##     ---
    ## 499996: 499996 -0.32  0.367  0  0 0
    ## 499997: 499997 -1.08  2.152  0  0 0
    ## 499998: 499998 -1.10  0.380  1  0 0
    ## 499999: 499999  0.56 -1.042  0  1 0
    ## 500000: 500000  0.52  0.076  0  1 1

The overall proportion of \(y=1\) in this case is

    dd[, mean(y)]

    ## [1] 0.56

If we have a desired marginal proportion of 0.40, then we can add an intercept of -0.66 to the data generating process:

\[ \text{logit}(y) = -0.66 + 0.15x_1 + 0.25x_2 + 0.10b_1 + 0.30b_2\]

The simulation now gives us the desired target:

    d1a <- defData(d1, varname = "y",
      formula = "t(c(-0.66, ..coefs1)) %*% c(1, x1, x2, b1, b2)",
      dist = "binary", link = "logit")

    genData(500000, d1a)[, mean(y)]

    ## [1] 0.4

If we change the distribution of the covariates, so that \(x_1 \sim N(1, 1)\), \(x_2 \sim N(2, 1)\), \(b_1 \sim Bin(0.5)\), and \(b_2 \sim Bin(0.8)\), and the strength of the association of these covariates with the outcome so that

\[ \text{logit}(y) = 0.20x_1 + 0.35x_2 + 0.20b_1 + 0.45b_2,\]

the marginal proportion/prevalence (assuming no intercept term) also changes, going from 0.56 to 0.84:

    coefs2 <- c(0.20, 0.35, 0.20, 0.45)

    d2 <- defData(varname = "x1", formula = 1, variance = 1)
    d2 <- defData(d2, varname = "x2", formula = 3, variance = 1)
    d2 <- defData(d2, varname = "b1", formula = 0.5, dist = "binary")
    d2 <- defData(d2, varname = "b2", formula = 0.8, dist = "binary")

    d2a <- defData(d2, varname = "y",
      formula = "t(..coefs2) %*% c(x1, x2, b1, b2)",
      dist = "binary", link = "logit")

    genData(500000, d2a)[, mean(y)]

    ## [1] 0.84

But under this new distribution, adding an intercept of -2.13 yields the desired target.

\[ \text{logit}(y) = -2.13 + 0.20x_1 + 0.35x_2 + 0.20b_1 + 0.45b_2 \]

    d2a <- defData(d2, varname = "y",
      formula = "t(c(-2.13, ..coefs2)) %*% c(1, x1, x2, b1, b2)",
      dist = "binary", link = "logit")

    genData(500000, d1a)[, mean(y)]

    ## [1] 0.4

#### Finding the intercept

Where did those two intercepts come from? The paper by Peter Austin describes an iterative bisection procedure that takes a distribution of covariates and a set of coefficients to identify the intercept coefficient that yields the target marginal proportion or prevalence.

The general idea of the algorithm is to try out series of different intercepts in an intelligent way that ends up at the right spot. (If you want the details for the algorithm, take a look at the paper.) The starting search range is pre-defined (we’ve used -10 to 10 for the intercept), and we start with an value of 0 for the initial intercept and simulate a large data set (the paper uses 1 million observations, but 100,000 seems to work just fine) and record the population prevalence. If we’ve overshot the target prevalence, we turn our attention to the range between -10 and 0, taking the average, which is -5. Otherwise, we focus on the range between 0 and 10. We iterate this way, choosing the range we need to focus on and setting the intercept at the mid-point (hence the name bisection). The algorithm will converge pretty quickly on the value of the intercept that gives the target population prevalence for the underlying covariate distribution and coefficient assumptions.

In the current implementation in `simstudy`, the intercept is provided by a simple call to `logisticCoefs`. Here are the calls for the two sets of definitions in definition tables d1 and d2.

    logisticCoefs(defCovar = d1, coefs = coefs1, popPrev = 0.40)

    ##    B0    x1    x2    b1    b2
    ## -0.66  0.15  0.25  0.10  0.30

    logisticCoefs(defCovar = d2, coefs = coefs2, popPrev = 0.40)

    ##    B0    x1    x2    b1    b2
    ## -2.13  0.20  0.35  0.20  0.45

### Risk ratios

Just as the prevalence depends on the distribution of covariates and their association with the outcome, risk ratios comparing the outcome probabilities for two groups also depend on the additional covariates. The marginal risk ratio comparing treatment (\(A =1\) to control (\(A=0\)) (given the distribution of covariates) is

\[RR = \frac{P(y=1 | A = 1)}{P(y=1 | A = 0)}\] In the data generation process we use a log-odds ratio of -0.40 (odds ratio of approximately 0.67) in both cases, but we get different risk ratios (0.82 vs. 0.93), depending on the covariates (defined in d1 and d2).

    d1a <- defData(d1, varname = "rx", formula = "1;1", dist = "trtAssign")
    d1a <- defData(d1a, varname = "y",
      formula = "t(c(-0.40, ..coefs1)) %*% c(rx, x1, x2, b1, b2)",
      dist = "binary", link = "logit"
    )

    dd <- genData(500000, d1a)
    dd[rx==1, mean(y)]/dd[rx==0, mean(y)]

    ## [1] 0.82

    d2a <- defData(d2, varname = "rx", formula = "1;1", dist = "trtAssign")
    d2a <- defData(d2a, varname = "y",
      formula = "t(c(-0.40, ..coefs2)) %*% c(rx, x1, x2, b1, b2)",
      dist = "binary", link = "logit"
    )

    dd <- genData(500000, d2a)
    dd[rx==1, mean(y)]/dd[rx==0, mean(y)]

    ## [1] 0.93

By specifying both a population prevalence and a target risk ratio in the call to `logisticCoefs`, we can get the necessary parameters. When specifying the target risk ratio, it is required to be between 0 and 1/popPrev. A risk ratio cannot be negative, and the probability of the outcome under treatment cannot exceed 1 (which will happen if the risk ratio is greater than 1/popPrev).

    C1 <- logisticCoefs(d1, coefs1, popPrev = 0.40, rr = 0.85)
    C1

    ##    B0     A    x1    x2    b1    b2
    ## -0.66 -0.26  0.15  0.25  0.10  0.30

If we use \(C_1\) in the data generation process, we will get a data set with the desired target prevalence and risk ratio:

    d1a <- defData(d1, varname = "rx", formula = "1;1", dist = "trtAssign")
    d1a <- defData(d1a, varname = "y",
      formula = "t(..C1) %*% c(1, rx, x1, x2, b1, b2)",
      dist = "binary", link = "logit"
    )

    dd <- genData(500000, d1a)

Here are the prevalence and risk ratio:

    dd[rx==0, mean(y)]

    ## [1] 0.4

    dd[rx==1, mean(y)]/dd[rx==0, mean(y)]

    ## [1] 0.86

You can do the same for the second set of assumptions.

### Risk differences

Risk differences have the same set of issues, and are handled in the same way. The risk difference is defined as

\[ RD = P(y=1 | A = 1) - P(y=1 | A = 0)\]

To get the coefficients related to a population prevalence of 0.40 and risk difference of -0.15 (so that the proportion in the exposure arm is 0.25), we use the rd argument:

    C1 <- logisticCoefs(d1, coefs1, popPrev = 0.40, rd = -0.15)
    C1

    ##    B0     A    x1    x2    b1    b2
    ## -0.66 -0.71  0.15  0.25  0.10  0.30

Again, using \(C_1\) in the data generation process, we will get a data set with the desired target prevalence and risk difference:

    d1a <- defData(d1, varname = "rx", formula = "1;1", dist = "trtAssign")
    d1a <- defData(d1a, varname = "y",
      formula = "t(..C1) %*% c(1, rx, x1, x2, b1, b2)",
      dist = "binary", link = "logit"
    )

    dd <- genData(500000, d1a)

    dd[rx==0, mean(y)]

    ## [1] 0.4

    dd[rx==1, mean(y)] - dd[rx==0, mean(y)]

    ## [1] -0.15

### AUC

The AUC is another commonly used statistic to evaluate a logistic model. We can use `logisticCoefs` to find the parameters that will allow us to generate data from a model with a specific AUC. To get the coefficients related to a population prevalence of 0.40 and an AUC of 0.85, we use the auc argument (which must be between 0.5 and 1):

    C1 <- logisticCoefs(d1, coefs1, popPrev = 0.40, auc = 0.85)
    C1

    ##    B0    x1    x2    b1    b2
    ## -1.99  0.85  1.41  0.56  1.69

Again, using \(C_1\) in the data generation process, we will get a data set with the desired target prevalence and the AUC (calculated here using the `lrm` function in the `rms` package:

    d1a <- defData(d1, varname = "y",
      formula = "t(..C1) %*% c(1, x1, x2, b1, b2)",
      dist = "binary", link = "logit"
    )

    dd <- genData(500000, d1a)

    dd[, mean(y)]

    ## [1] 0.4

    fit <- rms::lrm(y ~ x1 + x2 + b1 + b2, data = dd)
    fit$stats["C"]

    ##    C
    ## 0.85

References:

Austin, Peter C. “The iterative bisection procedure: a useful tool for determining parameter values in data-generating processes in Monte Carlo simulations.” BMC Medical Research Methodology 23, no. 1 (2023): 1-10.