Sample size considerations using mathematical models: an example with Chlamydia trachomatis infection and its sequelae pelvic inflammatory disease

Background The success of an intervention to prevent the complications of an infection is influenced by the natural history of the infection. Assumptions about the temporal relationship between infection and the development of sequelae can affect the predicted effect size of an intervention and the sample size calculation. This study investigates how a mathematical model can be used to inform sample size calculations for a randomised controlled trial (RCT) using the example of Chlamydia trachomatis infection and pelvic inflammatory disease (PID). Methods We used a compartmental model to imitate the structure of a published RCT. We considered three different processes for the timing of PID development, in relation to the initial C. trachomatis infection: immediate, constant throughout, or at the end of the infectious period. For each process we assumed that, of all women infected, the same fraction would develop PID in the absence of an intervention. We examined two sets of assumptions used to calculate the sample size in a published RCT that investigated the effect of chlamydia screening on PID incidence. We also investigated the influence of the natural history parameters of chlamydia on the required sample size. Results The assumed event rates and effect sizes used for the sample size calculation implicitly determined the temporal relationship between chlamydia infection and PID in the model. Even small changes in the assumed PID incidence and relative risk (RR) led to considerable differences in the hypothesised mechanism of PID development. The RR and the sample size needed per group also depend on the natural history parameters of chlamydia. Conclusions Mathematical modelling helps to understand the temporal relationship between an infection and its sequelae and can show how uncertainties about natural history parameters affect sample size calculations when planning a RCT. Electronic supplementary material The online version of this article (doi:10.1186/s12879-015-0953-5) contains supplementary material, which is available to authorized users.


Sample size considerations using mathematical models: an example with Chlamydia trachomatis infection and its sequelae pelvic inflammatory disease
Sereina A Herzog, Nicola Low, Andrea Berghold Additional File 1 List of Figures S1 Figure S1: List of Tables S1 Table S1: Change prevalence or duration of infection and fix the other parameter . . . 7 S2

Cumulative incidence of pelvic inflammatory disease (PID)
The function for the cumulative incidence of PID cases C(t) at time point t depends for each type of progression (immediate progression, constant progression, progression at the end) on the specific incidence of PID cases and fulfills that C(0) = 0. The equations are similar to Herzog et al. 1 where s = S(0), i 1 = I 1 (0) and i 2 = I 2 (0) describe the initial conditions for the constant progression scenario and satisfy the equation s + i 1 + i 2 = 1. The intervention group was treated for chlamydia infections and therefore has s = 1, i 1 = 0, and i 2 = 0. The control group starts at steady state in the absence of the intervention with s = 1 − p, i 1 = p r r+γ = p(1 − f ), and i 2 = p γ r+γ = pf . The initial conditions for immediate progression and progression at the end of infection (where γ = 0 and I(t) = I 1 (t) + I 2 (t)) are described as S(0) = s and I(0) = i 1 + i 2 .

Remark:
The cumulative PID incidence in the control group at time t equals for all three types of progression f rpt. This is a consequence of our assumption that independent of the type of progression a certain fraction f of all infected women will develop PID in the absence of an intervention. We had to set the progression rate γ = f r 1−f to achieve the same cumulative PID incidence in all three types of progression. The duration of infection in the constant progression type is 1 r : Total mean duration being infected = Mean duration in I 1 + Probability going from from I 1 to I 2 times mean duration in I 2 = 1 r + γ + P [I 1 to I 2 ] 1 r The relative risk (RR) at time point t > 0 differs for each type of progression (immediate progression, constant progression, progression at the end) being dependent on the specific cumulative incidence of PID cases for the intervention group.
RR(t) = cumulative PID incidence in the intervention group at time t cumulative PID incidence in the control group at time t The cumulative PID incidence in the control group is the same for all three types of progression (= f rpt), see section 1 on page 2. For the immediate progression and for the progression at the end, the RR is independent of the fraction f of infected women who develop PID: The RR depends on the fraction f for the constant progression because we set γ = f r 1−f to achieve the same PID incidence in all processes in absence of the intervention (i.e. in the control group).

Sample size and relative risk
Sample size needed per group is calculated using the method for the comparison of two proportions. 2 π 0 Proportion in control group who has the disease π 1 Proportion in intervention group who has the disease π π 0 +π 1 2 u One-sided percentage point of the normal distribution corresponding to 100% -power (e.g. Percentage point of normal distribution corresponding to the (two-sided) significance level (e.g. if α = 0.05 −→ v = 1.96) n Sample size per group In our calculations, π 0 equals the cumulative incidence of PID cases in the control group and π 1 equals the cumulative incidence of PID cases in the intervention group at time t (i.e. at follow-up time). Using the relative risk (RR) with RR = π 1 π 0 yields

The relation between sample size, relative risk and PID incidence while varying fraction developing PID
We investigated what happens to the sample size needed per group if fraction f of women who develop PID is increased or decreased, respectively.
The sample size needed per group depends on the PID incidence in the control group and the relative risk (RR) -see section 3: • The sample size needed per group is decreasing with increasing PID incidence in the control group and vice versa.
• The closer the RR is to 1 (i.e. no effect of intervention), the bigger is the sample size needed per group.
Note, the RR differs for the three types of progression (see section 2).

Immediate progression and progression at the end
For the immediate progression and for the progression at the end, the RR is independent of the fraction f of women who develop PID (see section 2). This means that changing fraction f influences the sample size needed per group only through the relationship between PID incidence and fraction f . The PID incidence in the control group is = f rpt.
For f −→ 0: The sample size needed per group is increasing with decreasing f because PID incidence in the control group is decreasing and (if everything else is kept constant).
For f −→ 1: The sample size needed per group is decreasing with increasing f because PID incidence in the control group is increasing (if everything else is kept constant).

Constant progression
The relationship between sample size needed per group and fraction f is more complex for the constant progression. The PID incidence as well as the RR of the constant progression (RR constant ) depends on the fraction f (see section 2).
With the properties of RR constant we can conclude that there exists a f * such that RR constant = 1 because RR immediate ≥ 1 and RR end ≤ RR immediate . Note, no analytical solution for f * can be derived.
The PID incidence in the control group and the RR constant act in opposing directions if we look at the sample size needed per group for The RR constant decreases with f −→ 0 and the effect size |1 − RR constant | increases, i.e. sample size needed would decrease but the PID incidence decreases with f −→ 0 and hence sample size needed would increase.
The RR constant increases with f −→ f * and the effect size |1 − RR constant | decreases, i.e. sample size needed would increase but the PID incidence increases with f −→ f * , i.e. sample size needed would decrease.
This considerations illustrate that there has to be a f # at which the sample size needed is minimised.
There are five distinction of cases (see Figure S1):  Figure S1: Influence of fraction f on sample size needed per group. There are five distinction of cases for the relationship between sample size needed per group and fraction f in the constant progression. The sample size is minimised at f # and the RR constant = 1 at f * .
• Case 1: For 0 ≤ f < f # , the influence of the PID incidence is stronger than the RR constant , i.e. sample size needed per group decreases for f −→ f # .
• Case 2: For f = f # , the sample size needed is minimised.
• Case 3: For f # < f < f * , the influence of the PID incidence is weaker than the RR constant , i.e. sample size needed per group increases for f −→ f * .
• Case 4: For f = f * , RR constant = 1 and no sample size needed per group can be calculated due to the division by 0. Note, in this situation we would expect no effect from the intervention.
• Case 5: For f * < f ≤ 1, the RR constant increases and is bigger than 1 for f −→ 1 and the effect size |1 − RR constant | increases, i.e. sample size needed decreases and also the PID incidence increases which results in a decrease in sample size needed per group for f −→ 1, i.e. sample size needed per group decreases.

Results of sensitivity analysis for scenario 1
In scenario 1 of the POPI trial, assuming a 2% incidence of PID, a sample size of 2,115 women per group would allow the investigators to detect a RR of 0.48 (80% power, 5% significance level) using the method for the comparison of two proportions (see section 3 on page 4). 3, 4 In the sensitivity analysis for scenario 1, the fraction f of women who develop PID in order to achieve the 2% PID incidence ranged between 15.9-80.3% at different levels of chlamydia prevalence and duration of infection, see Figure S2. The probability of multiple infections within the follow-up time decreases with increasing duration of infection, hence the fraction f has to increase in order to achieve the 2% PID incidence in the control group.   Figure S2: Sensitivity analysis for scenario 1 -fraction f. The fraction f of infected women who develop PID needed in order to achieve the 2% PID incidence in the control group.
For constant progression and progression at the end of the infection, changing the duration of infection influences the RR and the required sample size more than changing prevalence , see Table S1 and Figure S3. The hypothesis of immediate progression was not investigated because the estimated RR was > 1 in the main analysis .
For constant progression and progression at the end, Table S1 shows the range of the RR and the sample size needed per group while prevalence is kept constant and duration of infection is varied or vice versa.   The sample size calculated with constant progression is closer to the POPI trial sample size calculation (2115 women per group) than with progression at the end of infection in 78.5% of the investigated combinations of prevalence and duration of infection. Figure S4 shows the difference between the sample size calculated by constant progression and progression at the end relative to the POPI trial sample size calculation which uses 2% PID incidence with a RR=0.48. The difference ranged between -34.1% to 44.5%. A negative difference means that the sample size calculation from the constant progression is closer to the POPI trial sample size calculation than the sample size calculated by progression at the end. The sample size calculated with constant progression is closer to the POPI trial sample size calculation (2115 women per group) than with progression at the end of infection in 78.5% of the investigated combinations of prevalence and duration of infection. Figure S4 shows the difference between the sample size calculated by constant progression and progression at the end relative to the POPI trial sample size calculation which uses 2% PID incidence with a RR=0.48. The difference ranged between -34.1% to 44.5%. A negative difference means that the sample size calculation from the constant progression is closer to the POPI trial sample size calculation than the sample size calculated by progression at the end. 6 Generic sample size calculation using the mathematical model

Resulting RR and sample size for immediate progression varying duration of infection and fraction developing PID
The median RR is 1.046 (range 1.041-1.050) for the immediate progression model (Figure S5, Panel A) and the corresponding sample size needed per group has a median of 1,065,000 (range 812,000-1,589,000; Figure S5, Panel B).

Including an immunity stage -a SIRS model
There is an ongoing discussion about the existence and duration of immunity after a chlamydia infection. 5 We therefore investigated how our results for the sample size calculations used in the POPI trial would alter if we include an immunity stage. We used a Susceptible-Infected-Recovered-Susceptible (SIRS) compartmental model with which we investigated the same three hypothetical temporal relationship assumptions but having an immunity stage R included (see Figure S6). We denote with 1/δ the duration of immunity. therefore a decrease in the sample size needed per group. Note, a RR> 1 means that the predicted PID incidence in the intervention group is higher than in the control group.
In summary, including an immunity stage did not alter the results in our study about the sample size calculations used in the POPI trial: • Scenario 1, assuming that PID can develop throughout the infection period (constant progression) results in relative risk (RR) values between 0.491 and 0.498 including an immunity stage. This is compatible with the RR=0.48 assumed by the POPI trial investigators for their first sample size calculation.
• Scenario 2, assuming that PID develops at the end of infection results in RR values between 0.391 and 0.397 including an immunity stage. This is closest to the RR=0.44 assumed by the POPI trial investigators for their second sample size calculation. PID, pelvic inflammatory disease; RR, relative risk. Table S2: Scenario 1 with SIRS model. Analysing scenario 1 with 2% PID incidence in the control group and a relative risk (RR) of 0.48 using the SIRS model (80% power, 5% significance level). For the three types of progression we derived the PID incidence in the intervention group, the corresponding RR, and the sample size needed per group. Note, the first row with 0 years duration of immunity presents the results observed by the SIS model. in the intervention group (per year) in %. PID, pelvic inflammatory disease; RR, relative risk. Table S3: Scenario 2 with SIRS model. Analysing scenario 2 with 3% PID incidence in the control group and a relative risk (RR) of 0.44 using the SIRS model (80% power, 5% significance level). For the three types of progression we derived the PID incidence in the intervention group, the corresponding RR, and the sample size needed per group. Note, the first row with 0 years duration of immunity presents the results observed by the SIS model.