T-Distribution

T-Distribution in Small-Sample Health Economic Analysis

When a mean cost, a mean QALY gain or a regression coefficient is divided by a standard error worked out from the same patients, the ratio follows a t-distribution rather than the normal curve when the data are close to normal. The curve is indexed by its degrees of freedom, and with few of them its tails are heavier than those of the normal distribution, so the multiplier for a 95% interval is larger than 1.96. Health economists meet it in pilot studies and small trials, in t tests on mean costs, in extracting published results and in small-sample adjustments to meta-analysis. Using normal multipliers in its place makes intervals too narrow, and the understatement can pass into the input distributions of a decision model.

Student's problem and a trial of sleeping drugs

The distribution was published in Biometrika in 1908 in "The Probable Error of a Mean", signed only "Student". Ziliak explains that the author was William Sealy Gosset, a master brewer and experimental scientist at Guinness's brewery in Dublin, whose employer required his identity to be shielded from competitors. Student's problem is one that small health studies still face. As the number of experiments falls, he wrote, the standard deviation found from the sample becomes itself subject to an increasing error, until judgments based on the normal tables may become altogether misleading.

His first illustration was a drug comparison. Using data from Cushny and Peebles on two optical isomers of hyoscyamine hydrobromide, he analysed the differences in extra hours of sleep for 10 patients given both forms, which had mean 1.58 hours and standard deviation 1.17. He expressed the mean in units of a standard deviation calculated with divisor $n$, a ratio of 1.35, and read odds of about 666 to 1 that the laevo form was the better soporific. In modern notation the standard deviation with divisor $n-1$ is 1.23, the standard error is 0.389 and the t statistic is 4.06 on 9 degrees of freedom, which is Student's ratio multiplied by the square root of 9. The one-sided P value is 0.0014 and the 95% confidence interval for the mean gain is 0.70 to 2.46 hours.

Student also showed why the choice of curve matters. For samples of 10 and a gap of 1.3 sample standard deviations between sample and population means, the normal curve gave one-sided odds of about 7,000 to 1 where his curve gave about 550 to 1.

Building the distribution from a normal mean and an estimated spread

The modern definition expresses Student's ratio through two independent pieces. The numerator is standard normal, and the estimated scale in the denominator adds the extra uncertainty.

$$T = \frac{Z}{\sqrt{V/\nu}}$$

where $Z$ is a standard normal variable, $V$ is a chi-square variable with $\nu$ degrees of freedom independent of $Z$, and $\nu$ is the degrees of freedom of $T$. Student showed that for samples from a normal population there is no correlation between the sample mean and standard deviation, and for normal data the two are independent, as the construction needs. For one sample mean the statistic is

$$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$$

where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesised mean, $s$ is the sample standard deviation with divisor $n-1$, $n$ is the sample size and the degrees of freedom are $n-1$.

The NIST/SEMATECH e-Handbook gives the density in terms of the beta function.

$$f(x) = \frac{\left(1 + \frac{x^{2}}{\nu}\right)^{-(\nu+1)/2}}{B(0.5,\, 0.5\nu)\sqrt{\nu}}$$

where $x$ is the value of the variable, $\nu$ is the degrees of freedom and $B$ is the beta function. The handbook gives a mean of 0, undefined when $\nu$ is 1, and a standard deviation of $\sqrt{\nu/(\nu-2)}$, undefined when $\nu$ is 1 or 2. Its kurtosis is three times $(\nu-2)/(\nu-4)$ for $\nu$ above 4, so 9 at 5 degrees of freedom against 3 for the normal. With 1 degree of freedom the t is the Cauchy distribution, and the handbook calls the normal approximation quite good above 30.

The table, computed here, shows how fast the tails thin out. The first row is the two-sided 95% critical value and the second the probability of a value more than 3 units from zero.

With 4 degrees of freedom a value beyond 3 is about 15 times as likely as under the normal curve.

Where the t-distribution enters health economic analysis

Whenever a mean or a difference in means is standardised by a standard error estimated from the same data, the t-distribution supplies the reference for P values and the multiplier for intervals. Trial-based economic evaluation uses it for mean costs and QALYs per arm and for their differences, through the two-sample t test of hypothesis testing. In ordinary linear regression with normal errors, each coefficient divided by its standard error is referred to a t-distribution on the residual degrees of freedom, so the same logic covers adjusted cost differences.

Data extraction is a second use. Chapter 6 of the Cochrane Handbook says that when groups are small, say fewer than 60 participants each, confidence intervals should have been calculated from a t-distribution, and gives 2.0639 as the 95% value for a sample of 25. It also turns a reported t-test P value back into a t statistic on $N_E+N_C-2$ degrees of freedom, giving t = 2.78 for P = 0.008 with 45 degrees of freedom.

Evidence synthesis is a third. Chapter 10 of the Cochrane Handbook describes the Hartung-Knapp-Sidik-Jonkman adjustment for a random-effects model, which generally inflates the variance of the summary mean and uses a t-distribution with $k-1$ degrees of freedom, where $k$ is the number of studies. Its prediction interval uses the same distribution with the tau-squared estimate.

Assumptions behind t-based results for cost data

The exact t result assumes independent observations from a normal population. Student made that assumption explicitly, while judging that the deviation from normality must be very extreme to lead to serious error. Cost data test this, because they are usually skewed to the right by a few expensive patients.

Thompson and Barber argue that for trial costs, of the three tests they compare, only the t test on untransformed data addresses arithmetic means, the quantity needed to set a total budget, while a t test on log costs compares geometric means. They accept that t-based methods are strictly valid only for normal data but state that the test and its interval will be reliable if the skewness is not too extreme or the samples are moderately large. For small samples or extremely skewed data they recommend checking the results with the bootstrap, and they prefer methods that do not assume equal standard deviations in the two arms. That points to the Welch form of the test, whose degrees of freedom are never more than the pooled count of $N_1+N_2-2$.

Paired designs change the count again, since $n$ within-patient differences give $n-1$ degrees of freedom, 9 in Student's example rather than the 18 of two independent groups of 10. Student noted that the low standard deviation of those differences was probably due to the two drugs acting similarly on the same patient.

Worked example with community follow-up costs in a pilot trial

All numbers in this example are illustrative and were recomputed. A pilot trial randomises 10 patients to a nurse-led follow-up service and 10 to usual care, and records six-month costs.

1. Summarise each arm. The intervention arm has mean cost GBP 1,850 and standard deviation GBP 640, and usual care has mean GBP 1,420 and standard deviation GBP 380. The difference in mean cost is GBP 430.

2. Standard error. The variance terms are 640 squared over 10, or 40,960, and 380 squared over 10, or 14,440. They sum to 55,400, so the standard error is GBP 235.37. With equal arms the pooled-variance standard error is the same.

3. Degrees of freedom. The Welch-Satterthwaite formula divides 55,400 squared by the sum of 40,960 squared over 9 and 14,440 squared over 9, giving 14.64, against 18 for the pooled test.

4. Test statistic. The t statistic is 430 / 235.37 = 1.827. On 14.64 degrees of freedom the two-sided P value is 0.088, while the normal curve would give 0.068.

5. Interval. The 95% multiplier on 14.64 degrees of freedom is 2.136, so the half-width is 2.136 × 235.37 = 502.75 and the interval is minus GBP 73 to GBP 933. The normal half-width of 1.96 × 235.37 = 461.33 gives minus GBP 31 to GBP 891, about 8% narrower. The pooled test on 18 degrees of freedom gives a multiplier of 2.101 and minus GBP 65 to GBP 925.

6. Recover a standard error from a report. A separate trial with 15 patients per arm reports a mean utility gain of 0.060 with P = 0.04 from a two-sample t test. On 28 degrees of freedom the matching t statistic is 2.154, so the standard error is 0.060 / 2.154 = 0.0279. Since 2.048 × 0.0279 = 0.057, the 95% interval is 0.003 to 0.117. The normal value of 2.054 would give a standard error of 0.0292, about 5% too large.

Interpretation. The pilot is compatible with savings of about GBP 73 or extra spending of about GBP 933 per patient, and the normal curve would make the cost difference look more precise than the data allow. Step 6 errs the other way, because for a given P value the normal critical value is smaller than the t value, so dividing by it inflates the standard error.

The t-distribution as a modelling choice in Bayesian analysis

Gelman and colleagues, in Bayesian Data Analysis (section 3.2), show that for normal data under a noninformative prior on the mean and log standard deviation, the posterior distribution of the mean is a t-distribution on $n-1$ degrees of freedom, centred on the sample mean with scale $s/\sqrt{n}$. A mean input drawn from that t in probabilistic sensitivity analysis carries the uncertainty about the variance, which a normal input with the same standard error leaves out.

In section 17.2 the same authors describe the t as a longer-tailed replacement for the normal that can accommodate occasional unusual observations, or occasional extreme parameters in a prior distribution or hierarchical model. They read it as a mixture of normals with varying variances and warn that t-distributions with one or two degrees of freedom have infinite variance and are not usually realistic in the far tails.

Lee and Thompson replaced the usual normal random-effects distribution with the t and skewed extensions in examples including meta-analysis. Heavy tails and skewing mattered most for the predictive distribution, and inferences about the random effects could depend heavily on the choice, which bears on any Bayesian meta-analysis that feeds a model.

Computing t probabilities in Excel

Microsoft's help pages set out the spreadsheet functions. For the pilot, =T.INV.2T(0.05,14) returns the 95% multiplier, =T.DIST.2T(1.827,14) the two-sided P value and T.TEST with type 3 the unequal-variance test on raw costs. T.INV and T.INV.2T truncate a non-integer degrees of freedom argument, so a Welch value of 14.64 becomes 14, giving a multiplier of 2.145 and an interval of minus GBP 75 to GBP 935, slightly wider than the exact result. T.DIST.2T returns an error for a negative value.

Misreadings that recur

Most errors with the t-distribution either ignore it where it applies or expect more of it than it gives. Both kinds turn up in trial-based evaluations and model inputs.

1. Using 1.96 for a small sample. In the 20-patient pilot above, the normal interval is about 8% narrower than the t interval the data support.
2. Expecting the t to correct for skewness. It allows for an estimated variance, not for a non-normal shape, so very skewed small samples still need a check such as the bootstrap.
3. Assuming every t has a variance. With 2 or fewer degrees of freedom the variance is infinite, which matters when a t is used as a prior or an input distribution.
4. Converting reported P values with the normal curve. For a small trial this overstates the recovered standard error.
5. Equating a large t statistic with an important difference. Ziliak notes that Student took an economic approach and regarded statistical significance as nearly valueless in itself, which matches the decision focus of cost-effectiveness analysis.

Sources

• Student. The probable error of a mean. Biometrika. 1908;6(1):1-25. https://doi.org/10.2307/2331554
• Ziliak ST. Retrospectives: Guinnessometrics: the economic foundation of "Student's" t. Journal of Economic Perspectives. 2008;22(4):199-216. https://doi.org/10.1257/jep.22.4.199
• NIST/SEMATECH. e-Handbook of Statistical Methods. National Institute of Standards and Technology; 2012. Section 1.3.6.6.4. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm
• Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Sections 6.5.2.2 and 6.5.2.3. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06
• Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Sections 10.10.4.3 to 10.10.4.5. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-10
• Thompson SG, Barber JA. How should cost data in pragmatic randomised trials be analysed? BMJ. 2000;320(7243):1197-1200. https://doi.org/10.1136/bmj.320.7243.1197
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. CRC Press; 2013. Sections 3.2 and 17.2. http://www.stat.columbia.edu/~gelman/book/BDA3.pdf
• Lee KJ, Thompson SG. Flexible parametric models for random-effects distributions. Statistics in Medicine. 2008;27(3):418-434. https://doi.org/10.1002/sim.2897
• Microsoft. T.DIST, T.DIST.2T, T.INV, T.INV.2T and T.TEST function help pages. Microsoft Support; accessed 28 October 2026. https://support.microsoft.com/en-us/office/t-inv-2t-function-ce72ea19-ec6c-4be7-bed2-b9baf2264f17