Aim 1

Determine the effect of higher versus standard dose influenza vaccine on AKI

X-axis is the Odds Ratio (\(= \frac{p_S /(1 - p_S)}{p_H / (1 - p_H)}\)). Each curve assumes a proportion of AKI hospitalization over All hospitalization. Therefore, total AKI hospitalization rate (\(=\frac{p_S + p_H}{2}\)) will be calculated by:

With Odds Ratio and Total AKI hospitalization rate, we can solve for \(p_H\) and \(p_S\). Power is calculated by comparing two proportions.

Why this won’t work?

The main trial has 90% power to detect the proposed difference of composite endpoint using time to event analysis. First of all, we use 1 year event rate, which is smaller than what used in the main trial for three seasons. Secondly, we only look at AKI hospitalization, which is ~10% of the original event rate (70% * 15%). Therefore, our \(p_S + p_H\) is only 6% of \(p_S + p_H\) in the main trial. With OR=1.25, we are trying to detecting difference between \(p_H=0.0095\) and \(p_S=0.0119\) (while the main trial comparing 0.158 versus 0.198). Even with 8800 subjects (also smaller than original 9300), we still won’t have enough power to detect such a small difference.

Odds Ratio relation to Relative Risk

\[ RR \approx \frac{OR}{1-p_S+(p_S*OR)} \]

where \(p_S\) is small (<.01), so two measurements are aproximately equal (in our previous example 0.0119/0.0095 = 1.2526).

Aim 1 - revised

X-axis is the relative Risk Reduction for High Dose Vaccine (\(1 - p_H / p_S\)). Each curve assumes the rate of AKI complication among hospitalization in the standard dose group (\(p_S\)). Cohort is the 900 hospitalization from main trial.

Aim 2

Determine the effect of higher versus standard dose influenza vaccine on death/cardiopulmonary hospitalizations in patients with CKD

X-axis is the relative Risk Reduction for High Dose Vaccine (\(1 - p_H / p_S\)). Each curve assumes a proportion of CKD population in the whole cohort. Power is calculated by comparing two proportions.

Why Aim2 won’t work as well

CKD population is a fraction (~30%) of the population in the main trial. There are only two ways to power our significant smaller cohort:

  1. we have higher event rate, much higher than 9% per year.
  2. the effect size (relative risk, odds ratio etc) are much higher than proposed 20% difference. Which this 1/3 of the sample size, will require a relative risk of ~25%.

Otherwise, if our trial has enough power with same estimates and smaller sample size, the main trial would be an extremely over powered study; poor funding agency won’t allow that.

How to get enough power

I was thinking may be we could propose a composite endpoint of death, cardiopulmonary and AKI hospitalization with original cohort. Also, collect urine sample to identify CKD. Our primary analysis will be the same with main trial except that we are adding an extra element - AKI hospitalization to composite endpoint. This will increase overall event rate, but the influence on the effect size is not clear (might enlarge or attenuate the proposed difference in the main trial). And secondarey analysis would be in the subgroup of CKD. Here is an example of how the power looks.

Aim 3.1

Higher vs standard is associated with greater frequency of adequate immune response, as assessed by doubling of geometric mean antibody titers at 4 weeks

##    p.s    0.3   0.35    0.4   0.45    0.5   0.55    0.6
## 1 0.15 0.7223 0.9103 0.9813 0.9976 0.9998 1.0000 1.0000
## 2 0.20 0.3712 0.6633 0.8757 0.9701 0.9956 0.9996 1.0000
## 3 0.25 0.1239 0.3377 0.6212 0.8480 0.9600 0.9936 0.9994

Row: response rate for standard dose group, 0.15, 0.2, 0.25;

Column: response rate for high dose group, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6.

Aim 3.3

Influenza vaccination is associated with the development of anti-HLA antibodies

##    p.s   0.06   0.08    0.1   0.12   0.14
## 1 0.03 0.1752 0.3405 0.5193 0.6783 0.8011
## 2 0.04 0.0992 0.2209 0.3825 0.5504 0.6981
## 3 0.05 0.0610 0.1377 0.2679 0.4261 0.5843

Row: response rate for standard dose group, 0.03, 0.04, 0.05;

Column: response rate for high dose group, 0.06, 0.08, 0.1, 0.12, 0.14.