Reference to Relative Risk Cutofs For Statistical Signifiance

By Milo Schield, Augsburg College

If the relative risk is greater than one, the smallest value that will be statistically-significant occurs when the lower-limit of the 95% confidence interval for a relative risk just touches unity (or when the lower limit of the 95% confidence interval for the natural log of the relative risk just touches zero).

Setting Equation 3:

Equal to zero with the negative sign gives the minimum relative risk that is statistically significant. This relative risk cutoff is denoted by RRss.

Equation 4:

LN(RRss) - Z*Sqrt{(1/N)[(1-P1)/P1 + (1-RRssP1)/(RRssP1)]} = 0

Equation 5:

LN(RRss) = Z*Sqrt{(1/n)[(1-P1)/P1 + (1-RRssP1)/(RRssP1)]} Unfortunately RRss is on both sides of this equation. We are unaware of an analytic solution. An alternate approach is iterative. Start with a solution that is close and then iterate. Consider how RRss might be eliminated on the right side of Equation 5. Return to Equation 3. Note that

Equation 6:

if RR > 1 then P2 > P1 and (1-P2)/P2 < (1-P1)/P1.

Equation 7:

Sqrt[(2/n)(1-P1)/P1)] > Sqrt[(1/N)[ (1-P1)/P1 + (1-P2)/P2)] Inserting this into Equation 5 gives:

Equation 8:

LN(RRss) = Z*Sqrt[(2/N)(1-P1)/(P1*N1)]

Equation 9:

RRss = EXP{Z*Sqrt[(2/N)(1-P1)/(P1*N1)]}

Calculated Relative-Risk Cutoffs for Statistical Significance of Risk Ratios using Schield’s Equation 9 translated in R code.

# variable n is the number of participants in Hartmann-Boyce (2018)
n <- (31722 + 32918)

# variable n1 denotes the larger participant or control group in the study
n1 <- 32918

# variable p1 is the proportion of the smaller group (participant vs. control) in the study
p1 <- 31722 / n

# variable Z is the Z-score for a standard 95% cutoff 
Z <- 1.96

# This is equation 9 from Schiled
rr_equation <- Z * sqrt( (2/n)*(1-p1)/(p1 *n1))

# This is constant e 
e <- 2.718282

# This is the calculated cutoff, when output above is exponentiated using e
cutOff <- e ^ rr_equation

# Function to calculate the cutoff level of significance for the RR using the Shield equation
# using different parameters

cutOff_RR <- function(n, n1, p1, Z) {
  rr_equation <- Z * sqrt( (2/n)*(1-p1)/(p1 *n1))
  e <- 2.718282
  cutOff <- e ^ rr_equation
  result <-cat("The calculated cutoff level of signifcance for the RR is", cutOff)
  return(result)
}

# This is the calculated level of significance using for the pooled RR 
cat("The calculated cutoff level of significance for the RR is", cutOff)
## The calculated cutoff level of significance for the RR is 1.000061

The printed number is the level of significance needed to be significance in the pooled RR. Since the actual RR is 1.55 and above the cutoff value calculated, we can conclude the RR in Hartmann-Boyce (2018) NRTs for tobacco cessation is significant.