F-Distribution

F-Distribution in Health Economic Analysis

The F distribution is the reference distribution for tests that compare two independent estimates of variance, and health economists meet it mainly when analysing patient-level cost and outcome data. An F statistic appears in the output of almost every linear regression, including the cost regressions that sit behind a trial-based cost-effectiveness analysis, and in the first-stage diagnostics of instrumental variable studies. Its shape is set by two degrees of freedom, one for the numerator and one for the denominator. In analysis of variance and regression only large values of the ratio count as evidence against the null hypothesis. It is rarely an input to a decision model, so its job in health economics is judging evidence in data.

The F distribution as a ratio of variance estimates

The distribution is built from two independent chi-square variables. The NIST/SEMATECH e-Handbook of Statistical Methods describes the F distribution as the ratio of two chi-square distributions, each of which has first been divided by its degrees of freedom.

$$F = \frac{U_1 / \nu_1}{U_2 / \nu_2}$$

where $U_1$ and $U_2$ are independent chi-square variables, $\nu_1$ is the numerator degrees of freedom, $\nu_2$ is the denominator degrees of freedom and $F$ is the resulting ratio, which takes values of zero or more.

Each chi-square variable divided by its degrees of freedom has an expected value of one, so the ratio sits near one when both parts estimate the same variance. The handbook gives the mean as $\nu_2 / (\nu_2-2)$ when $\nu_2$ exceeds 2, which is 117/115 = 1.017 for the denominator degrees of freedom in the worked example below. The distribution is skewed to the right, most strongly with few degrees of freedom, and its percentage points have no simple closed form, so software computes them numerically.

Fisher, in Statistical Methods for Research Workers (1925), framed the same test through a quantity z, half the difference of the logarithms of the two mean squares, and tabulated its 5 per cent points. The variance ratio equals $e^{2z}$, so the two forms carry the same information.

F tests in cost regressions and analysis of variance

Mihaylova and colleagues note that methods based on the normal distribution, including the t test and ordinary least squares regression, are widely used to estimate mean costs. The F tests attached to these methods share their assumptions, and the NIST handbook states the one-way analysis of variance model with errors that are normally and independently distributed with mean zero and a common variance.

For a comparison of mean costs across several trial arms, the analysis of variance statistic divides the between-arm mean square by the within-arm mean square.

$$F = \frac{SSB / (k-1)}{SSW / (N-k)} = \frac{MSB}{MSW}$$

where $SSB$ is the between-group sum of squares, $SSW$ is the within-group sum of squares, $k$ is the number of groups, $N$ is the total number of patients, and $MSB$ and $MSW$ are the between-group and within-group mean squares. Under the null hypothesis of equal means the statistic follows an F distribution with $k-1$ and $N-k$ degrees of freedom. The NIST handbook explains that under the null hypothesis both mean squares estimate the error variance, so their ratio should be close to 1.

The same distribution serves for nested comparisons in multiple regression. A cost model may be fitted with and without a set of terms, such as dummies for recruiting centre or a group of treatment by subgroup interactions in a subgroup analysis, and the F test asks whether the set jointly reduces the residual sum of squares by more than chance would.

$$F = \frac{(SSR_r-SSR_u) / q}{SSR_u / (N-p)}$$

where $SSR_r$ is the residual sum of squares of the restricted model without the extra terms, $SSR_u$ is the residual sum of squares of the unrestricted model, $q$ is the number of restrictions tested, $N$ is the number of observations and $p$ is the number of parameters in the unrestricted model including the intercept. With normal errors the statistic follows an F distribution with $q$ and $N-p$ degrees of freedom. The analysis of variance test is the special case in which the restricted model contains only an intercept and the unrestricted model adds the arm dummies.

Instrument strength in observational studies

Instrumental variable analysis is one approach to uncontrolled confounding in observational comparative effectiveness research. In two-stage least squares the first stage regresses the treatment on the instruments. Brookhart, Rassen and Schneeweiss state that standard instrumental variable estimators have a finite sample bias inversely proportional to the F statistic that tests whether the instruments make a significant contribution to the first-stage model, and that the bias is towards the ordinary least squares estimate.

Brookhart and colleagues recommend reporting the first-stage F statistic and the partial r squared as a means of judging whether the instruments are strong enough, and record the rule of thumb that F statistics below 10 are potentially problematic with two-stage least squares and multiple instruments. Burgess and Thompson, studying genetic instruments, found that bias increases as the expected F statistic falls, called the rule of thumb that F above 10 avoids bias misleading, and warned that choosing instruments or models after seeing measured F statistics can make the bias worse. A first-stage F statistic is therefore a diagnostic of weakness, and a value above 10 is no certificate of an unbiased estimate.

F tests in meta-regression

Evidence synthesis for decision models may explore between-study heterogeneity with meta-regression, and the F distribution enters through small-sample adjustments. The documentation for the rma.uni function in the R package metafor states that, with the t option, tests of single coefficients use a t distribution with $k-p$ degrees of freedom while the omnibus test of $m$ coefficients uses an F distribution with $m$ and $k-p$ degrees of freedom, where $k$ is the number of estimates and $p$ the number of model coefficients. It explains that treating the estimated between-study variance as known gives standard errors that tend to be too small, which the Knapp and Hartung adjustment counters, and that adjustment also uses t and F distributions.

Worked example with 12-month costs in a three-arm trial

All numbers in this example are illustrative. A trial randomises 40 patients each to usual care, nurse-led follow-up and a digital monitoring service, and records 12-month health service costs. Mean costs are GBP 2,100, GBP 2,400 and GBP 2,850, with standard deviations of GBP 900, GBP 1,000 and GBP 1,100.

1. Between-arm mean square. With equal arms the grand mean is (2,100 + 2,400 + 2,850) / 3 = GBP 2,450. The deviations from it are minus 350, minus 50 and 400, and their squares sum to 122,500 + 2,500 + 160,000 = 285,000. Weighting by arm size gives $SSB$ = 40 × 285,000 = 11,400,000, and with 2 degrees of freedom $MSB$ = 11,400,000 / 2 = 5,700,000.

2. Within-arm mean square. The three variances sum to 810,000 + 1,000,000 + 1,210,000 = 3,020,000, and each arm contributes 39 times its variance, so $SSW$ = 39 × 3,020,000 = 117,780,000. With 117 degrees of freedom (120 patients less 3 arms), $MSW$ = 117,780,000 / 117 = 1,006,667.

3. F statistic and p-value. The ratio is 5,700,000 / 1,006,667 = 5.66 on 2 and 117 degrees of freedom. The 5% critical value is 3.07 and the right-tail probability is 0.0045, so the hypothesis of equal mean costs in the three arms is rejected at the 5% level.

4. Two arms and the t test. For usual care against the digital service, the pooled variance is (810,000 + 1,210,000) / 2 = 1,010,000, and the standard error of the difference is the square root of 50,500, or 224.72. The t statistic is 750 / 224.72 = 3.34. The two-arm analysis of variance gives $SSB$ = 40 × (375 squared + 375 squared) = 11,250,000 and F = 11,250,000 / 1,010,000 = 11.14, which equals the square of the unrounded t statistic, 3.3375. Both tests give a p-value of 0.0013 on 78 denominator degrees of freedom.

5. Nested test for baseline covariates. The arm-only regression of cost has a residual sum of squares equal to $SSW$, 117,780,000. Adding four baseline covariates, such as age and prior-year admissions, gives an unrestricted model with 7 parameters and an illustrative residual sum of squares of 98,000,000. The residual sum of squares falls by 19,780,000, and 19,780,000 / 4 = 4,945,000. The unrestricted residual mean square is 98,000,000 / 113 = 867,257, so F = 4,945,000 / 867,257 = 5.70 on 4 and 113 degrees of freedom, with a p-value of 0.0003.

6. Interpretation. The tests show that mean costs differ between arms and that the covariates explain part of the variation in cost. They do not settle the decision. Claxton argued that decisions should rest on mean net benefit whatever the statistical significance of the differences, with the spread of net benefit relevant to whether more information is needed. The GBP 750 difference between the digital service and usual care would enter the economic evaluation alongside the difference in QALYs, and a non-significant F test would not justify treating the arms as equal in cost.

Links with the t and chi-square distributions

Squaring a t distribution variable with $\nu$ degrees of freedom gives an F variable with 1 and $\nu$ degrees of freedom, so a two-sided t test and the matching F test always agree, as step 4 shows. Fisher noted that his z table for one numerator degree of freedom held the logarithms of the 5 per cent values of the t table. In the example the two-sided 5% t critical value on 78 degrees of freedom is 1.9908, and its square, 3.963, is the 5% critical value of F with 1 and 78 degrees of freedom.

As the denominator degrees of freedom grow, $\nu_1 F$ approaches a chi-square variable with $\nu_1$ degrees of freedom, which Fisher also observed for an infinite second sample. Hence software reports some joint tests as chi-square Wald tests and others as F tests. For two restrictions the large-sample 5% critical value of F is 5.991 / 2 = 2.996, compared with 3.074 on 117 denominator degrees of freedom and 4.103 on 10. The F version is exact only for the normal linear model, so generalised linear models of cost, such as a gamma model with a log link, usually rely on chi-square Wald or likelihood ratio tests.

Computing F probabilities in Excel

Microsoft documents F.DIST.RT as the probability that an F variable exceeds a given value, and F.INV.RT as returning critical values when the significance level is the probability argument. Non-integer degrees of freedom are truncated.

The NIST handbook warns that its upper critical value notation is the opposite of the designation used by some texts and software, so the tail should always be checked.

Assumptions and common misreadings

Most errors with the F distribution in health economics come from forgetting what the test assumes or from reading a significance result as a decision. The first concerns cost data in particular.

1. Ignoring the skewness of costs. Mihaylova and colleagues report that normal-based methods give unbiased estimates for randomised data but are sensitive to extreme values and likely to be inefficient in small to medium samples when costs are not normal. With large samples the central limit theorem supports inference on means, and bootstrapping offers a check.
2. Testing variances with a normality-dependent test. The NIST handbook says Bartlett's test is sensitive to departures from normality and may simply be testing for non-normality, and recommends the median-based Levene test, whose statistic is referred to an F distribution with $k-1$ and $N-k$ degrees of freedom, as performing well against many types of non-normal data.
3. Treating F above 10 as proof that weak instrument bias is absent. Burgess and Thompson call that rule of thumb misleading, and the F statistic says nothing about instrument validity.
4. Reading significance as cost-effectiveness. An F test answers whether means differ, not which option offers more net monetary benefit.
5. Using the F distribution as a model input. The NIST handbook notes that it is typically used for hypothesis tests and confidence intervals and rarely for modelling, and probabilistic sensitivity analysis draws its inputs from other distributions such as the beta and gamma.

Sources

• Brookhart MA, Rassen JA, Schneeweiss S. Instrumental variable methods in comparative safety and effectiveness research. Pharmacoepidemiology and Drug Safety. 2010;19(6):537-554. https://doi.org/10.1002/pds.1908
• Burgess S, Thompson SG; CRP CHD Genetics Collaboration. Avoiding bias from weak instruments in Mendelian randomization studies. International Journal of Epidemiology. 2011;40(3):755-764. https://doi.org/10.1093/ije/dyr036
• Claxton K. The irrelevance of inference: a decision-making approach to the stochastic evaluation of health care technologies. Journal of Health Economics. 1999;18(3):341-364. https://doi.org/10.1016/S0167-6296(98)00039-3
• Fisher RA. Statistical Methods for Research Workers. Oliver and Boyd; 1925. Chapter 7, Intraclass correlations and the analysis of variance. http://psychclassics.yorku.ca/Fisher/Methods/chap7.htm
• Microsoft. F.DIST.RT function; F.INV.RT function. Microsoft Support, accessed 25 October 2026. https://support.microsoft.com/en-us/office/f-dist-rt-function-d74cbb00-6017-4ac9-b7d7-6049badc0520 ; https://support.microsoft.com/en-us/office/f-inv-rt-function-d371aa8f-b0b1-40ef-9cc2-496f0693ac00
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• NIST/SEMATECH. e-Handbook of Statistical Methods. Sections 1.3.5.7, 1.3.5.9, 1.3.5.10, 1.3.6.6.5, 7.4.3.2 and 7.4.3.3, accessed 25 October 2026. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3665.htm
• Viechtbauer W. rma.uni, meta-analysis via linear (mixed-effects) models. metafor package documentation, accessed 25 October 2026. https://wviechtb.github.io/metafor/reference/rma.uni.html