supplemental material

(1)

Rethinking Page 1 of 4 v5

SUPPLEMENTAL MATERIAL Statistical Analysis Methods Effectiveness Matrix

Table S1. The effectiveness matrix for the condom and PrEP use categories of never, sometimes, and

always. Condom categories are never (0%), sometimes (1-99%), and always (100%). PrEP categories are never (0%), sometimes (<50% use), and always (≥90% use).

PrEP Use

Never (N) Sometimes (S) Always (A) Never (N)

Sometimes (S) Condom Use

Always (A)

The effectiveness for the i^th row (condom use) and j^th column (PrEP use) is given by

, where RR is the risk ratio or hazard ratio for condom and PrEP with “never” as the reference category. Our estimation for the effectiveness matrix assumes that PrEP and condom use effectiveness are independent. Under this assumption we have the following estimates for the effectiveness matrix.

, ,

(2)

, , Incidence Matrix

Table S2. The incidence matrix for the condom and PrEP use categories of never, sometimes, and

always. Condom categories are never (0%), sometimes (1-99%), and always (100%). PrEP categories are never (0%), sometimes (<50% use), and always (≥90% use).

PrEP Use

Never (N) Sometimes (S) Always (A) Never (N)

Sometimes (S) Condom Use

Always (A)

The incidence for the i^th row (condom use) and j^th column (PrEP use) is given by

, where RR is the risk ratio or hazard ratio for condom and PrEP use with “never” as the reference category and I is the estimated incidence in the absence of PrEP or condom use (category of never). Our estimation for the incidence matrix assumes that PrEP and condom use RR are independent.

In addition, we assume the RR is independent of the baseline I that’s estimated in the absence of PrEP and condom use. Under this assumption we have the following estimates for the incidence matrix.

, ,

,

(3)

, ,

To estimate the variance we take the natural-log of the estimated incidence, . This

results in . It follows, assuming independence of condom and

PrEP use that the variance is estimated as

The 95% confidence interval (CI) of the estimated incidence on the natural-log is given by

Once we have our estimated incidence matrix we can multiply these estimates by a given population to estimate our incidence and corresponding 95% CI. As an example of the computations we illustrate the estimate for the incidence and corresponding 95% CI using the condom and PrEP use groups “always”.

The estimate on the log-scale is given by:

Exponentiation leads to an estimate of the incidence

The 95% CI is estimated as

Substituting our estimated variances results in

(4)

Hence, our incidence 95% CI estimates (0.0031, 0.0245) are obtain by