Mortality –> prevalence conversion in logit space

#####
# what i think you're doing currently...

# -- estimate mortality/prevalence ratios from observed data
set.seed(123)

df <- data.frame(
  mort1 = runif(n = 1000, min = 0.03, max = 0.05), 
  prev1 = runif(n = 1000, min = 0.19, max = 0.21)
)
df$mp_ratio <- df$mort1 / df$prev1
df$mp_ratio_logit <- qlogis(df$mp_ratio)

fit1 <- lm(mp_ratio_logit ~ 1, data = df)
plogis(fit1$coefficients) # gives a MP ratio of about 0.2

## (Intercept) 
##   0.1983003

# -- apply the predicted ratio to mortality to get an estimate of prevalence

df_pred <- data.frame(mort2 = seq(0.02, 0.24, by = 0.02))
df_pred$logit_mp_pred <- predict(fit1, newdata = df_pred)
df_pred$prev2 <- df_pred$mort2 / plogis(df_pred$logit_mp_pred) # uh oh, some prevalence are >1

df_pred

##    mort2 logit_mp_pred     prev2
## 1   0.02     -1.396952 0.1008571
## 2   0.04     -1.396952 0.2017143
## 3   0.06     -1.396952 0.3025714
## 4   0.08     -1.396952 0.4034285
## 5   0.10     -1.396952 0.5042857
## 6   0.12     -1.396952 0.6051428
## 7   0.14     -1.396952 0.7059999
## 8   0.16     -1.396952 0.8068571
## 9   0.18     -1.396952 0.9077142
## 10  0.20     -1.396952 1.0085713
## 11  0.22     -1.396952 1.1094285
## 12  0.24     -1.396952 1.2102856

#####
# what i'd suggest
# -- this will work as long as mortality is always >0 and <1

df$logit_diff <- qlogis(df$mort1) - qlogis(df$prev1)
fit2 <- lm(logit_diff ~ 1, data = df)

#-- the predictions are now answering the question,
#   how much do i need to adjust a logit-space mortality observations to 
#.  turn it into logit-space prevalence
#-- it's no longer interpretable as a MP ratio

df_pred$logitscale_adjustment_pred <- predict(fit2, newdata = df_pred)
df_pred$prev3 <- plogis(qlogis(df_pred$mort2) - df_pred$logitscale_adjustment_pred)

df_pred

##    mort2 logit_mp_pred     prev2 logitscale_adjustment_pred     prev3
## 1   0.02     -1.396952 0.1008571                  -1.803096 0.1101976
## 2   0.04     -1.396952 0.2017143                  -1.803096 0.2018200
## 3   0.06     -1.396952 0.3025714                  -1.803096 0.2791988
## 4   0.08     -1.396952 0.4034285                  -1.803096 0.3454159
## 5   0.10     -1.396952 0.5042857                  -1.803096 0.4027238
## 6   0.12     -1.396952 0.6051428                  -1.803096 0.4528074
## 7   0.14     -1.396952 0.7059999                  -1.803096 0.4969515
## 8   0.16     -1.396952 0.8068571                  -1.803096 0.5361538
## 9   0.18     -1.396952 0.9077142                  -1.803096 0.5711999
## 10  0.20     -1.396952 1.0085713                  -1.803096 0.6027177
## 11  0.22     -1.396952 1.1094285                  -1.803096 0.6312143
## 12  0.24     -1.396952 1.2102856                  -1.803096 0.6571043