The total number of actual events (e.g. hospital-acquired infections) reported is compared to the predicted number derived from the reference database.

With this methodology, the SIR for the national baseline is 1.0. However, the standardized ratio is only a point estimate, and needs additional information to indicate if it is statistically significantly. Confidence intervals (95%) are used to determine if the difference has meaning or if the variation may be due to change.

SIR is calculated by the indirect method of standardization. The confidence interval is derived from the Poisson distribution. The hypothesis test (z-test) is used to calculate P values, but the data being compared must be normally distributed. When the sample size is greater than 30, this will be the case, and can be confirmed by a minimum expected cell frequency greater than 1.0. When it is less than 1.0, Fisher's Exact Test (based on the hypergeometric distribution) is used, although the SIR is usually not calculated in this case because it does not meet the minimum requirement for precision. Reporting on too few procedures is a risk to patient confidentiality, so if there are fifty or fewer central line days, the report for the number of infections is usually deferred until there are more central line days. The lower bound of the confidence interval is calculated only if the infection count > 0.

The ratio of two standardized ratios (for example, comparing results for this year and last, or for comparing two surgeons) is the ratio of the two SIR values calculated from risk stratified data (Poisson distribution), but the confidence interval in this case is derived from the binomial distribution.

Relevant EXCEL functions [ Excel worksheet {sir_excel.xls} ]

• Normal distribution: probability (p) from z value
  • p = 1-NORMSDIST(z)
• Poisson distribution: probability (p) for observed (O_x) and expected (E_x) counts
  • P = MIN[POISSON(O_x, E_x, TRUE), 1-POISSON(O_x-1, E_x, TRUE)]
• Poisson distribution: confidence interval from count (O_x) (where α = confidence as decimal eg 0.95)
  • Lower Limit = CHIINV[(1+α)/2, 2*O_x]*0.5
  • Upper Limit = CHIINV[(1-α)/2, 2*O_x+2]*0.5
• Binomial distribution: confidence interval for proportions (where x=sample, n=population (denominator), α = confidence as decimal eg 0.05)
  • Method (a):
    Lower Limit = x/[x+(n-x+1)*FINV(α/2, 2*(n-x+1), 2*x)]
    Upper Limit = (x+1)*FINV(α/2, 2*(x+1), 2*(n-x))/[n-x+(x+1)*FINV(α/2, 2*(x+1), 2*(n-x))]
  • Method (b): gives the same result as method (a)
    Lower Limit = 1-BETAINV(1-α/2, n-x+1, x)
    Upper Limit = 1-BETAINV(α/2, n-x, x+1)
• Binomial distribution : probability (p) for proportions
  • x=sample (numerator), n=denominator, p=α (eg 0.05 for 95% confidence interval):
    p = MIN[BINOMDIST(x, n, p, TRUE), 1-BINOMDIST(x-1, n, p, TRUE)]
  • When comparing two standardised ratios: probability (p) for observed (O_x, O_y) with expected counts (E_x, E_y)
    p = MIN[BINOMDIST(x, n, p, TRUE), 1-BINOMDIST(x-1, n, p, TRUE)]
    where x = O_x, n = O_x+O_y, p = E_y/(E_x+E_y)
• Fisher Exact Test (Excel for small values, otherwise use EpiInfo or other statistical software)
  Sample {i, n} and NNIS {I, N} ⇒ i,I = number of SSI; n,N = denominators; x=loop counter.
  For x=0 To 9
      a = AND[x<=(i+I), x≤n]
      b = IF(a, HYPGEOMDIST[x, n, (i+I), (n+N)], "")
      c = HYPGEOMDIST[x, n, I, N]
      d = IF(b<c*1.00001, b, "")
  Next
  P = Σ(d)

Table 1: Culver ^[3] page 2
Data for the sample to be assessed are shown in blue (Columns C and D).
The NNIS population data (shown in red in Columns F and G) should be obtained from the latest published NHSN report ^[8] .
You should complete Table 1 yourself using the Excel equations as shown in Columns E and Column H, and cells C7, D7, and E7.

Table 3: Culver ^[3] page 13
Data continued from Table 1, with columns C to H hidden for convenience on this webpage.
Columns I to L were added as follows:
Column I: P {stands for proportion} ` P = (i+I)/(n+N) `
I2 = (C2+F2)/(D2+G2)
copy this formula downwards from I2 to I6.
first row => I2 = (3+103)/(80+5088) = 0.02051

Column J: ` J* = (1/n + 1/N) `
J2 = (1/D2+1/G2)
copy this formula downwards from J2 to J6.
first row => J2 = (1/80+1/5088) = 0.01270

Rates r and R are the two proportions that we wish to compare. Now calculate the following z-statistic (formula 1, page 6).
K column: `z = (|r-R| - 0.5 times (1/n+1/N))/sqrt(P times (1-P) times (1/n+1/N)) `

`K2 = (ABS(E2 - H2) - 0.5 times J2)/sqrt(I2 times (1-I2) times J2)`
copy this formula downwards from K2 to K6.

A SIR of 1.0 means that a hospital is doing the same as the reference database (national average), a SIR of less than 1.0 means there were fewer than expected, and a SIR above 1.0 means there were more.

• If the 95% confidence interval range for a particular hospital includes the value 1.0 (lower limit less than 1.0, upper limit greater than 1.0), the actual SIR value is not statistically different from the national average.
• If the range does not include 1.0 then the actual value is statistically significantly different from the national average. However, a low rate may be due to under-reporting of infections, or to infrequent and/or short duration of device use.

Example Report

Number of operations: N = 524
Number of SSI: O = 13
Infection rate: r = 13 ÷ 524 = 0.02481 (2.481%)
NHSN reference rate: R = 74 ÷ 5,640 = 0.01276 (1.276%)
Expected number of SSI: E = (0.01276*524) = 6.687

SIR = O/E = 13 ÷ 6.687 = 1.944

Confidence interval for SIR
[a] First calculate 95% confidence interval from Poisson distribution (O=13, α=0.95):
… Lower Limit(O_L) = CHIINV[(1+0.95)/2, 2*13]*0.5 = 6.922
… Upper Limit(O_U)= CHIINV[(1-0.95)/2, 2*13+2]*0.5 = 22.23

[b] Then calculate 95% confidence interval for SIR:
… Lower Limit= (O_L/E) = (6.922/6.687) = 1.035
… Upper Limit= (O_U/E) = (22.23/6.687) = 3.324
⇒ Confidence interval for SIR: 1.944 (1.035~3.324)

P (O=13, E=6.687)
= MIN[ POISSON(13, 6.687,TRUE), 1-POISSON(13-1, 6.687,TRUE) ]
= 0.01963764 {P<0.05}

Practice

Calculate the SIR, p-value, and 95% confidence interval for each of the following.

[a] the observed number of SSIs that occurred (22) to the expected number (13.6) … in Table 4. Infection Control Report — Team A on p.20 in reference [Culver DH ³]

[b] observed 64 cases of leukaemia in children … and calculated that 45.6 cases would be expected. on p.1315 of reference [Morris MJ ²]

Solutions

[a]
SIR = O/E = 22/13.6 = 1.62
P = MIN[ POISSON(22, 13.6,TRUE), 1-POISSON(22-1, 13.3,TRUE) ] = 0.02195
O_L = CHIINV[(1+0.95)/2, 2*22]*0.5 = 13.78728
O_U = CHIINV[(1-0.95)/2, 2*22+2]*0.5 = 33.30826
95%CI_L = O_L / E = 13.78728 / 13.6 = 1.01377
95%CI_U = O_U / E = 33.30826 / 13.6 = 2.44914
Result: SIR ⇒ 1.62 (1.01~2.45) P=0.022

[b]
SIR = O/E = 64/45.6 = 1.4035
P = MIN[ POISSON(64, 45.6,TRUE), 1-POISSON(64-1, 45.6,TRUE) ] = 0.0058
O_L = CHIINV[(1+0.95)/2, 2*64]*0.5 = 49.2878
O_U = CHIINV[(1-0.95)/2, 2*64+2]*0.5 = 81.7266
95%CI_L = O_L / E = 49.2878 / 45.6 = 1.0809
95%CI_U = O_U / E = 81.7266 / 45.6 = 1.7922
Result: SIR ⇒ 1.40 (1.08~1.79) P=0.006

he Ratio of Two Standardised Ratios

Using Example 5 on page 28 of the reference compare the SIRs of two orthopedic surgeons. [Culver DH ³]

SIR_x = O_x / E_x = 8 / 8.44 = 0.95
SIR_y = O_y / E_y = 8 / 2.60 = 3.08
SIR_x ÷ SIR_y = 0.95 / 3.08 = 0.308

Using the nomenclature in [Morris & Gardner ²] (A_L, A_U, B_L, B_U) and the binomial distribution:
A_L
= x/(x+(n-x+1)*FINV(α/2, 2*(n-x+1), 2*x))
= O_x/(O_x+([O_x+O_y]-O_x+1)*FINV(α/2, 2*([O_x+O_y]-O_x+1), 2*O_x))
= 8/(8+([8+8]-8+1)*FINV(0.05/2, 2*([8+8]-8+1), 2*8))
= 0.2465
B_L
= A_L/(1-A_L)
= 0.2465/(1-0.2465)
= 0.32716

A_U
= (x+1)*FINV(α/2, 2*(x+1), 2*(n-x))/(n-x+(x+1)*FINV(α/2, 2*(x+1), 2*(n-x)))
= (O_x+1)*FINV(α/2, 2*(O_x+1), 2*([O_x+O_y]-O_x))/([O_x+O_y]-O_x+(O_x+1)*FINV(α/2, 2*(O_x+1), 2*([O_x+O_y]-O_x)))
= (8+1)*FINV(0.05/2, 2*(8+1), 2*([8+8]-8))/([8+8]-8+(8+1)*FINV(0.05/2, 2*(8+1), 2*([8+8]-8)))
= 0.7535
B_U
= A_U/(1-A_U)
= 0.7535/(1-0.7535)
= 3.0566

Confidence interval of the two standardised ratios
= B_L*(E_y/E_x)
= 0.32716 * (2.60/8.44)
= 0.10078

= B_U*(E_y/E_x)
= 3.0566 * (2.60/8.44)
= 0.94161

P (binomial)
= MIN(BINOMDIST(x, n, p, TRUE), 1-BINOMDIST(x-1, n, p, TRUE)) P (binomial)
= MIN(BINOMDIST(x, n, p, TRUE), 1-BINOMDIST(x-1, n, p, TRUE))
= MIN(BINOMDIST(O_x, [O_x+O_y], E_y/(E_x+E_y), TRUE), 1-BINOMDIST(O_x-1, [O_x+O_y], E_y/(E_x+E_y), TRUE))
= MIN(BINOMDIST(8, [8+8], 2.60/(8.44+2.60), TRUE), 1-BINOMDIST(8-1, [8+8], 2.60/(8.44+2.60), TRUE))
= 0.01910 (P<0.05)

Results: comparison of surgeon X to surgeon Y
The ratio of the two SIRs: 0.308 (0.10~0.94), P=0.019

Practice

Calculate the SIR, p-value, and 95% confidence interval for the following.

[a] in the section Ratio of Two Standardised Ratios on p.1315 of reference [Morris JA, Garnder MJ ²]
O₁=64, E₁=45.6, O₂=25, E₁=23.7

Solutions

SIR₁ = O₁ / E₁ = 64 / 45.6 = 1.4035
SIR₂ = O₂ / E₂ = 25 / 23.7 = 1.0549
SIR₁ ÷ SIR₂ = 1.4035 / 1.0549 = 1.3305

Using the nomenclature in [Morris & Gardner ²] (A_L, A_U, B_L, B_U) and the binomial distribution:
A_L
= x/(x+(n-x+1)*FINV(α/2, 2*(n-x+1), 2*x))
= O_x/(O_x+([O_x+O_y]-O_x+1)*FINV(α/2, 2*([O_x+O_y]-O_x+1), 2*O_x))
= 64/(64+([64+25]-64+1)*FINV(0.05/2, 2*([64+25]-64+1), 2*64))
= 0.6138
B_L
= A_L/(1-A_L)
= 0.6138/(1-0.6138)
= 1.5896

A_U
= (x+1)*FINV(α/2, 2*(x+1), 2*(n-x))/(n-x+(x+1)*FINV(α/2, 2*(x+1), 2*(n-x)))
= (O_x+1)*FINV(α/2, 2*(O_x+1), 2*([O_x+O_y]-O_x))/([O_x+O_y]-O_x+(O_x+1)*FINV(α/2, 2*(O_x+1), 2*([O_x+O_y]-O_x)))
= (64+1)*FINV(0.05/2, 2*(64+1), 2*([64+25]-64))/([64+25]-64+(64+1)*FINV(0.05/2, 2*(64+1), 2*([64+25]-64)))
= 0.8093
B_U
= A_U/(1-A_U)
= 0.8093/(1-0.8093)
= 4.2429

Confidence interval of the two standardised ratios
= B_L*(E_y/E_x)
= 1.5896 * (23.7/45.6)
= 0.8262

= B_U*(E_y/E_x)
= 4.2429 * (23.7/45.6)
= 2.2052

P (binomial)
= MIN(BINOMDIST(x, n, p, TRUE), 1-BINOMDIST(x-1, n, p, TRUE)) P (二項次)
= MIN(BINOMDIST(x, n, p, TRUE), 1-BINOMDIST(x-1, n, p, TRUE))
= MIN(BINOMDIST(O_x, [O_x+O_y], E_y/(E_x+E_y), TRUE), 1-BINOMDIST(O_x-1, [O_x+O_y], E_y/(E_x+E_y), TRUE))
= MIN(BINOMDIST(64, [64+25], 23.7/(45.6+23.7), TRUE), 1-BINOMDIST(64-1, [64+25], 23.7/(45.6+23.7), TRUE))
= 4.5408E-13

Results: the 95% confidence interval for the ratio of the two standardised ratios:
= 1.3305 (0.083~2.205), P<0.001

Compare each risk category of procedure with the NNIS standard

Using Table 1 on page 2 of reference [Culver DH ³]

[1] Using NHSN data as the reference database, get the control group infection rate for each risk category
[ NNIS Risk Index ]   

[2] Compare SSI rates for each risk category
Pooled proportion ("p-hat"): p̂ = (i + I) / (n + N)
Yates continuity correction = 0.5 * (1/n + 1/N)

z = [ |r-R| - 0.5*(1/n+1/N) ] / SQRT[ p̂*(1-p̂)*(1/n+1/N) ]
P = 1-NORMSDIST(z)

For example, first row (CARD-0,1)
z = [ |r-R| - 0.5*(1/n+1/N) ] / SQRT[ p̂*(1-p̂)*(1/n+1/N) ]
z = [ ABS(0.375-0.0202) - 0.5*(1/80+1/5088) ] / SQRT[ 0.020511*(1-0.020511)*(1/80+1/5088) ]
z = 0.682986 ⇒ P = 1-NORMSDIST(z)
P = 1-NORMSDIST(0.682986)
P = 0.247308

second row (CARD-2,3)
z = [ |r-R| - 0.5*(1/n+1/N) ] / SQRT[ p̂*(1-p̂)*(1/n+1/N) ]
z = [ ABS(0.15-0.052897) - 0.5*(1/20+1/1191) ] / SQRT[ 0.0545*(1-0.0545)*(1/20+1/1191) ]
z = 1.400513 ⇒ P = 1-NORMSDIST(z)
P = 1-NORMSDIST(1.400513)
P = 0.08068

CBGB-0
z = [ |r-R| - 0.5*(1/n+1/N) ] / SQRT[ p̂*(1-p̂)*(1/n+1/N) ]
z = [ ABS(0.1-0.015873) - 0.5*(1/10+1/819) ] / SQRT[ 0.01689*(1-0.01689)*(1/10+1/819) ]
z = 0.817589 ⇒ P = 1-NORMSDIST(z)
P = 1-NORMSDIST(0.817589)
P = 0.206796

CBGB-1
z = [ |r-R| - 0.5*(1/n+1/N) ] / SQRT[ p̂*(1-p̂)*(1/n+1/N) ]
z = [ ABS(0.043478-0.031499) - 0.5*(1/230+1/32065) ] / SQRT[ 0.031584*(1-0.031584)*(1/230+1/32065) ]
z = 0.845944 ⇒ P = 1-NORMSDIST(z)
P = 1-NORMSDIST(0.848944)
P = 0.198792

CBGB-2,3
z = [ |r-R| - 0.5*(1/n+1/N) ] / SQRT[ p̂*(1-p̂)*(1/n+1/N) ]
z = [ ABS(0.0833-0.057586) - 0.5*(1/60+1/7745) ] / SQRT[ 0.057783*(1-0.057783)*(1/60+1/7745) ]
z = 0.573746 ⇒ P = 1-NORMSDIST(z)
P = 1-NORMSDIST(0.573746)
P = 0.28307

None of the p-values is lower than the arbitrary cut point of 0.05, so none of the SSI rates are "significantly greater" than the NNIS rates.

SIR as a risk-adjusted summary measure of different risk categories.

Continuing Example 3 and calculating the expected number (E) of SSI and the SIR for each risk category
The Standardized Infection Ratio (SIR) is the ratio of observed number of SSI to the expected number.

Device utilization ratios are proportions, even though both the numerator and denominator involve the counting of patient-days.

Expected number of CLABSI: E = nR = (100,000)*(0.002) = 200

SIR_(ICU) = i/E = 170/200 = 0.85
95%CI: LCL ~ UCL
     = [CHIINV((1+α)/2,2*i)*0.5]/E ~ [CHIINV((1-α)/2,2*i+2)*0.5]/E
     = [CHIINV((1+0.95)/2, 2*170)*0.5]/200 ~ [CHIINV((1-0.95)/2, 2*170+2)*0.5]/200
     = 0.727 ~ 0.988

P = MIN(POISSON(i, E, TRUE), 1-POISSON(i-1, E, TRUE))
     = MIN(POISSON(170, 200,TRUE), 1-POISSON(170-1, 200, TRUE))
     = 0.017

Expected number of CLABSI: E = nR = (58,000)*(0.0015) = 87

SIR₍₅₉₁₃₎ = i/E = 58/87 = 0.667
95%CI: LCL ~ UCL
     = [CHIINV((1+α)/2,2*i)*0.5]/E ~ [CHIINV((1-α)/2,2*i+2)*0.5]/E
     = [CHIINV((1+0.95)/2, 2*58)*0.5]/87 ~ [CHIINV((1-0.95)/2, 2*58+2)*0.5]/87
     = 0.506 ~ 0.862

P = MIN(POISSON(i, E, TRUE), 1-POISSON(i-1, E, TRUE))
     = MIN(POISSON(58, 87, TRUE), 1-POISSON(58-1, 87, TRUE))
     = 0.000616

SIR as a summary measure for device utilization.

SIR_(ICU+5913) = Σi/ΣE = (i₁+i₂)/(E₁+E₂)
     = (170+58)/(200+87) = 228/287 = 0.794
95%CI: LCL ~ UCL
     = [CHIINV((1+α)/2,2*(i₁+i₂))*0.5]/(E₁+E₂) ~ [CHIINV((1-α)/2,2*(i₁+i₂)+2)*0.5]/(E₁+E₂)
     = [CHIINV((1+0.95)/2, 2*228)*0.5]/287 ~ [CHIINV((1-0.95)/2, 2*228+2)*0.5]/287
     = 0.695 ~ 0.905

P = MIN(POISSON((i₁+i₂), (E₁+E₂), TRUE), 1-POISSON((i₁+i₂)-1, (E₁+E₂), TRUE))
     = MIN(POISSON(228, 287, TRUE), 1-POISSON(228-1, 287, TRUE))
     = 0.000178