### EBM Stats Calculator

Note: This stats calculator requires JavaScript. Please ensure that you are using an up-to-date browser with JavaScript turned on. If you have technical difficulties, please contact david.newton@utoronto.ca.

The CEBM Statistics Calculator was created for your own personal use and testing purposes. It is to be used as a guide only. Medical decisions should NOT be based solely on the results of this program. Although this program has been tested thoroughly, the accuracy of the information cannot be guaranteed.

## CEBM Statistics Calculator

 A B C D LR+ LR-

### Results

Chi-squared — p-value: — — — — — — — — — — — — — — — — — — — — — — — ### Help

#### Getting Started

To perform a calculation, please choose a table type from the menu.

##### Table Type Options
Diagnostic Test
calculates the Sensitivity, Specificity, Positive Predictive Value (PPV), Negative Predictive Value (NPV), Likelihood Ratio + (LR+), and Likelihood Ratio – (LR-)
Prospective Study
calculates the Relative Risk (RR), Absolute Relative Risk (ARR), and Number Needed to Treat (NNT)
Case-control Study
calculates the Odds Ratio (OR)
Randomized Control Trial (RCT)
calculates the Relative Risk Reduction (RRR), Absolute Relative Risk (ARR), and Number Needed to Treat (NNT)

#### Equations

nr1 = a+b
nr2 = c+d
nc1 = a+c
nc2 = b+d
N = a + b + c + d
z = 1.959964

##### Diagnostic Test1,2,3
###### Sensitivity

Sensitivity = a/nc1
Lower limit = ((2×a) + z2 – z√( ( 4×a×c / nc1 ) + z2 )) / ((2×nc1) + (2×z2))
Upper limit = ((2×a) + z2 + z√( ( 4×a×c / nc1 ) + z2 )) / ((2×nc1) + (2×z2))

###### Specificity

Specificity = d/nc2
Lower limit = ((2×d) + z2 – z√( ( 4×d×b / nc2 ) + z2 )) / ((2×nc2) + (2×z2))
Upper limit = ((2×d) + z2 + z√( ( 4×d×b / nc2 ) + z2 )) / ((2×nc2) + (2×z2))

###### Positive Predictive Value (PPV)

PPV = a/nr1
Lower limit = ((2×a) + z2 – z√( ( 4×a×b / nr1 ) + z2 )) / ((2×nr1) + (2×z2))
Upper limit = ((2×a) + z2 + z√( ( 4×a×b / nr1 ) + z2 )) / ((2×nr1) + (2×z2))

###### Negative Predictive Value (NPV)

NPV = d/nr2
Lower limit = ((2×d) + z2 – z√( ( 4×d×c / nr2 ) + z2 )) / ((2×nr2) + (2×z2))
Upper limit = ((2×d) + z2 + z√( ( 4×d×c / nr2 ) + z2 )) / ((2×nr2) + (2×z2))

###### Likelihood Ratio +

LR+ = Sensitivity / (1 – Specificity)
Lower limit = exp ( ln( (nc2×a)/(nc1×b) ) – z√( ( c/(a×nc1) ) + ( d/(b×nc2) ) ) )
Upper limit = exp ( ln( (nc2×a)/(nc1×b) ) + z√( ( c/(a×nc1)) + ( d/(b×nc2) ) ) )

###### Likelihood Ratio –

LR+ = (1 – Sensitivity) / Specificity
Lower limit = exp ( ln( (nc2×c)/(nc1×d) ) – z√( ( a/(c×nc1) ) + ( b/(d×nc2) ) ) )
Upper limit = exp ( ln( (nc2×c)/(nc1×d) ) + z√( ( a/(c×nc1)) + ( b/(d×nc2) ) ) )

###### Pre-test Probability by Post-test Probability Plot

Plot PretestProb by PosttestProb(+) and PretestProb by PosttestProb(-).
PretestProb = a vector of values from 0 to 1.0 by 0.001 increments
PosttestProb(+) = (PretestOdds × LR+) / (1 + (PretestOdds × LR+))
PosttestProb(-) = (PretestOdds × LR-) / (1 + (PretestOdds × LR-))
where…
PretestOdds = PretestProb / (1 – PretestProb)

##### Prospective Study4, 5, 6
###### Chi-squared

Chi-squared = (N × ( | (a×d) – (b×c) | – (N / 2))2) / (nr1 × nr2 × nc1 × nc2)
p-value = Chi-squared density at 1 degree of freedom

###### Relative Risk (RR)

RR = (a × (c+d)) / (c × (a+b))
Lower limit = exp ( ln(RR) – z√( (1/c) – (1/(c+d)) + (1/a) – (1/(a+b)) )
Upper limit = exp ( ln(RR) + z√( (1/c) – (1/(c+d)) + (1/a) – (1/(a+b)) )

###### Absolute Risk Reduction (ARR)

ARR = (c/(c+d)) – (a/(a+b))
Lower limit = ARR – z√( (u2×(1-u2) / (a+b)) + (w1×(1-w1) / (c+d)) )
Upper limit = ARR + z√( (u1×(1-u1) / (c+d)) + (w2×(1-w2) / (a+b)) )
where…
u1 = ((2×c) + z2 + z√( ( 4×c×d / nr2 ) + z2 )) / ((2×nr2) + (2×z2))
u2 = ((2×a) + z2 + z√( ( 4×a×b / nr1 ) + z2 )) / ((2×nr1) + (2×z2))
w1 = ((2×c) + z2 – z√( ( 4×c×d / nr2 ) + z2 )) / ((2×nr2) + (2×z2))
w2 = ((2×a) + z2 – z√( ( 4×a×b / nr1 ) + z2 )) / ((2×nr1) + (2×z2))

###### Number Needed to Treat (NNT)

NNT = 1/ARR (rounded down to 0 decimal places)
Lower limit = 1 / ARR lower limit (rounded to 0 decimal places)
Upper limit = 1 / ARR upper limit (rounded to 0 decimal places)

##### Case-Control Study4,7
###### Chi-squared

Chi-squared = (N × ( | (a×d) – (b×c) | – (N / 2))2) / (nr1 × nr2 × nc1 × nc2)
p-value = Chi-squared density at 1 degree of freedom

###### Odds Ratio (OR)

OR = (a × d) / (b × c)
Lower limit = exp ( ln(OR) – z√( (1/a) + (1/b) + (1/c) + (1/d) )
Upper limit = exp ( ln(OR) + z√( (1/a) + (1/b) + (1/c) + (1/d) )

##### Randomized Controlled Trial (RCT)4,6
###### Chi-squared

Chi-squared = (N × ( | (a×d) – (b×c) | – (N / 2))2) / (nr1 × nr2 × nc1 × nc2)
p-value = Chi-squared density at 1 degree of freedom

###### Relative Risk Reduction (RRR)

RRR = ((c / (c+d)) – (a / (a+b))) / (c / (c+d))
Lower limit = 1 – ( exp( ln( (a × (c+d)) / (c × (a+b)) ) + z√( (1/c) – (1/(c+d)) + (1/a) – (1/(a+b)) ) ) )
Upper limit = 1 – ( exp( ln( (a × (c+d)) / (c × (a+b)) ) – z√( (1/c) – (1/(c+d)) + (1/a) – (1/(a+b)) ) ) )

###### Absolute Risk Reduction (ARR)

ARR = (c/(c+d)) – (a/(a+b))
Lower limit = ARR – z√( (u2×(1-u2) / (a+b)) + (w1×(1-w1) / (c+d)) )
Upper limit = ARR + z√( (u1×(1-u1) / (c+d)) + (w2×(1-w2) / (a+b)) )
where…
u1 = ((2×c) + z2 + z√( ( 4×c×d / nr2 ) + z2 )) / ((2×nr2) + (2×z2))
u2 = ((2×a) + z2 + z√( ( 4×a×b / nr1 ) + z2 )) / ((2×nr1) + (2×z2))
w1 = ((2×c) + z2 – z√( ( 4×c×d / nr2 ) + z2 )) / ((2×nr2) + (2×z2))
w2 = ((2×a) + z2 – z√( ( 4×a×b / nr1 ) + z2 )) / ((2×nr1) + (2×z2))

###### Number Needed to Treat (NNT)

NNT = 1/ARR (rounded down to 0 decimal places)
Lower limit = 1 / ARR lower limit (rounded to 0 decimal places)
Upper limit = 1 / ARR upper limit (rounded to 0 decimal places)

##### References

For full references, see the Credits & References section.

1. Wilson (1927)
95% confidence interval for sensitivity, specificity, PPV and NPV
2. Newcombe (1998a)
95% confidence interval for sensitivity, specificity, PPV and NPV
3. Simel et al (1991)
95% confidence interval for LR+ and LR-
4. Yates (1934)
Chi-squared
5. Armitage & Berry (1994)
95% confidence interval for RR
6. Newcombe (1998b)
95% confidence interval for ARR
7. Bland & Altman (2000)
95% confidence interval for OR

#### Troubleshooting

If you encounter an error, it is most likely because invalid (i.e. non-numeric) data was entered, or because the data you entered forced an equation to attempt to divide by zero. In either case, try modifying your input.

If you continue to have problems, please contact David Newton at david.newton@utoronto.ca

#### Contact

If you are having trouble with the calculator, please contact David Newton at david.newton@utoronto.ca

#### Credits

CEBM Statistics Calculator © 2004-2010 CEBM.

Statisticians
Farah Khandwala, Kevin Thorpe
Developer
David Newton
Based on the Java CEBM Statistics Calculator by
Peter Wong
SAJAX Simple AJAX Toolkit by
MODERNMETHOD; used under the BSD license
The Probability Distribution Library (PDL) (a descendant of the JSci Project source tree) by
Meagher et al; used under the GNU Lesser General Public License

#### References

1. Wilson, E. B. “Probable inference, the law of succession, and statistical inference,” Journal of the American Statistical Association 22: 209-212, 1927.
2. Newcombe, RG. “Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods,” Statistics in Medicine17: 857-872, 1998 (a).
3. Simel DL, Samsa GP, Matchar DB. “Likelihood ratios with confidence: sample size estimation for diagnostic test studies,” Journal of Clinical Epidemiology 44: 763-70, 1991.
4. Yates, F. “Contingency table involving small numbers and the Χ2 test,” Journal of the Royal Statistical Society (Supplement) 1: 217-235, 1934.
5. Armitage P and Berry G. Statistical Methods in Medical Research (3rd ed.), Blackwell, 1994.
6. Newcombe RG. “Interval estimation for the difference between independent proportions: Comparison of eleven methods,” Statistics in Medicine 17: 873-890, 1998 (b).
7. Bland JM and Altman DG. “The odds ratio,” British Medical Journal Statistics Notes 320: 1468, 2000.