The model is of the form
\[ P(Y = 1) = \lambda(1 - logit^{-1}(\beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_kX_k)) \] Where
\(Y\) - binary outcome indicator, (eg. 1 - infected, 0 - not infected).
\(X\) - covariate.
\(k\) - number of covariates.
Model fitting is done with a function called sclr. It is used in the same way as other fitting functions like lm.
# One-titre fit to included simulated data
fit1 <- sclr(status ~ logHI, one_titre_data)
summary(fit1)
#> Call: status ~ logHI
#>
#> Parameter estimates
#> lambda beta_0 beta_logHI
#> 0.4912562 -5.4253573 2.1487774
#>
#> 95% confidence intervals
#> 2.5 % 97.5 %
#> lambda 0.4662442 0.5162682
#> beta_0 -6.4417802 -4.4089345
#> beta_logHI 1.8146909 2.4828639
#>
#> Log likelihood: -2469.765
# Two-titre fit to included simulated data
fit2 <- sclr(status ~ logHI + logNI, two_titre_data)
summary(fit2)
#> Call: status ~ logHI + logNI
#>
#> Parameter estimates
#> lambda beta_0 beta_logHI beta_logNI
#> 0.4692433 -5.6536038 2.2098460 1.0409995
#>
#> 95% confidence intervals
#> 2.5 % 97.5 %
#> lambda 0.4402525 0.4982341
#> beta_0 -6.6252070 -4.6820006
#> beta_logHI 1.8981507 2.5215412
#> beta_logNI 0.8738325 1.2081665
#>
#> Log likelihood: -1858.46The predict method will return the point estimate and a confidence interval of \(\beta_0 + \beta_1X_1 + ... + \beta_kX_k\) where \(k\) is the number of covariates. It will also apply the inverse logit transformation to these estimates and interval bounds to get the point estimate and the interval for the probability of protection on the original scale.
# One-titre fit
preddata1 <- data.frame(logHI = seq(0, 8, length.out = 101))
pred1 <- predict(fit1, preddata1)
head(pred1[, c("logHI", "prot_l", "prot_point", "prot_u")])
#> # A tibble: 6 x 4
#> logHI prot_l prot_point prot_u
#> <dbl> <dbl> <dbl> <dbl>
#> 1 0 0.00159 0.00438 0.0120
#> 2 0.08 0.00194 0.00520 0.0139
#> 3 0.16 0.00236 0.00617 0.0160
#> 4 0.24 0.00288 0.00732 0.0185
#> 5 0.32 0.00351 0.00868 0.0213
#> 6 0.4 0.00427 0.0103 0.0246
# Two-titre fit
preddata2 <- data.frame(logHI = seq(0, 8, length.out = 101), logNI = 1)
pred2 <- predict(fit2, preddata2)
head(pred2[, c("logHI", "logNI", "prot_l", "prot_point", "prot_u")])
#> # A tibble: 6 x 5
#> logHI logNI prot_l prot_point prot_u
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 0 1 0.00432 0.00983 0.0222
#> 2 0.08 1 0.00527 0.0117 0.0258
#> 3 0.16 1 0.00643 0.0139 0.0299
#> 4 0.24 1 0.00785 0.0166 0.0347
#> 5 0.32 1 0.00957 0.0197 0.0402
#> 6 0.4 1 0.0117 0.0235 0.0466To get the estimated titre (and the confidence interval) that corresponds to a particular protection level (eg. 50%), use the get_protection_level function. Its interface is similar to that of predict.
protHI1 <- get_protection_level(fit1, "logHI", lvl = 0.5)
print(protHI1)
#> # A tibble: 3 x 3
#> logHI prot_prob est
#> <dbl> <dbl> <chr>
#> 1 2.39 0.5 low bound
#> 2 2.52 0.5 point
#> 3 2.64 0.5 upper bound
prot_lvls2 <- data.frame(logNI = log(c(0.1, 10, 40)))
protHI2 <- get_protection_level(fit2, "logHI", prot_lvls2)
print(protHI2)
#> logNI logHI prot_prob est
#> 1 -2.302585 3.4024635 0.5 low bound
#> 2 2.302585 1.3380153 0.5 low bound
#> 3 3.688879 0.6568784 0.5 low bound
#> 4 -2.302585 3.6430566 0.5 point
#> 5 2.302585 1.4736837 0.5 point
#> 6 3.688879 0.8206373 0.5 point
#> 7 -2.302585 3.8723188 0.5 upper bound
#> 8 2.302585 1.5883861 0.5 upper bound
#> 9 3.688879 0.9605298 0.5 upper bound