Multiple Comparisons and Outliers
Dunnett's Test Table: Critical Values for Comparisons with a Control
Critical values of Dunnett's t for comparing several treatment groups with one control group while controlling the familywise error rate.
- k = 1 to 12 treatments
- One- and two-sided
- Exact calculator
Dunnett's Test Table
Jump to a table below. Rows: error df · Columns: number of treatment groups compared with the control (k) · Cells: critical |t|
Two-sided test, α = 0.05
| Treatment groups compared with control (k) → df ↓ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 2.571 | 3.030 | 3.293 | 3.476 | 3.615 | 3.727 | 3.821 | 3.900 | 3.970 | 4.032 | 4.087 | 4.137 |
| 6 | 2.447 | 2.863 | 3.099 | 3.263 | 3.388 | 3.489 | 3.573 | 3.644 | 3.707 | 3.763 | 3.812 | 3.857 |
| 7 | 2.365 | 2.752 | 2.971 | 3.123 | 3.238 | 3.331 | 3.408 | 3.475 | 3.533 | 3.584 | 3.630 | 3.671 |
| 8 | 2.306 | 2.673 | 2.880 | 3.023 | 3.131 | 3.219 | 3.292 | 3.354 | 3.408 | 3.457 | 3.500 | 3.539 |
| 9 | 2.262 | 2.614 | 2.812 | 2.948 | 3.052 | 3.135 | 3.205 | 3.264 | 3.316 | 3.362 | 3.403 | 3.440 |
| 10 | 2.228 | 2.568 | 2.759 | 2.890 | 2.990 | 3.070 | 3.137 | 3.194 | 3.244 | 3.288 | 3.328 | 3.364 |
| 11 | 2.201 | 2.532 | 2.717 | 2.845 | 2.941 | 3.019 | 3.084 | 3.139 | 3.187 | 3.230 | 3.268 | 3.303 |
| 12 | 2.179 | 2.502 | 2.683 | 2.807 | 2.901 | 2.977 | 3.040 | 3.094 | 3.140 | 3.182 | 3.219 | 3.253 |
| 13 | 2.160 | 2.478 | 2.654 | 2.776 | 2.868 | 2.942 | 3.003 | 3.056 | 3.102 | 3.142 | 3.179 | 3.212 |
| 14 | 2.145 | 2.457 | 2.631 | 2.750 | 2.840 | 2.912 | 2.973 | 3.024 | 3.069 | 3.109 | 3.144 | 3.177 |
| 15 | 2.131 | 2.439 | 2.610 | 2.727 | 2.816 | 2.887 | 2.946 | 2.997 | 3.041 | 3.080 | 3.115 | 3.147 |
| 16 | 2.120 | 2.424 | 2.592 | 2.708 | 2.795 | 2.865 | 2.924 | 2.974 | 3.017 | 3.056 | 3.090 | 3.121 |
| 17 | 2.110 | 2.410 | 2.577 | 2.691 | 2.777 | 2.846 | 2.904 | 2.953 | 2.996 | 3.034 | 3.068 | 3.099 |
| 18 | 2.101 | 2.399 | 2.563 | 2.676 | 2.761 | 2.830 | 2.887 | 2.935 | 2.977 | 3.015 | 3.048 | 3.079 |
| 19 | 2.093 | 2.388 | 2.551 | 2.663 | 2.747 | 2.815 | 2.871 | 2.919 | 2.961 | 2.998 | 3.031 | 3.061 |
| 20 | 2.086 | 2.379 | 2.540 | 2.651 | 2.735 | 2.802 | 2.857 | 2.905 | 2.946 | 2.983 | 3.016 | 3.045 |
| 24 | 2.064 | 2.349 | 2.507 | 2.614 | 2.695 | 2.760 | 2.814 | 2.860 | 2.900 | 2.936 | 2.968 | 2.996 |
| 30 | 2.042 | 2.321 | 2.474 | 2.578 | 2.657 | 2.720 | 2.772 | 2.817 | 2.856 | 2.890 | 2.921 | 2.949 |
| 40 | 2.021 | 2.293 | 2.441 | 2.543 | 2.619 | 2.680 | 2.731 | 2.774 | 2.812 | 2.845 | 2.875 | 2.902 |
| 60 | 2.000 | 2.265 | 2.410 | 2.508 | 2.582 | 2.642 | 2.691 | 2.733 | 2.769 | 2.801 | 2.830 | 2.856 |
| 120 | 1.980 | 2.238 | 2.379 | 2.475 | 2.547 | 2.604 | 2.651 | 2.692 | 2.727 | 2.758 | 2.786 | 2.811 |
| ∞ | 1.960 | 2.212 | 2.349 | 2.442 | 2.511 | 2.567 | 2.613 | 2.652 | 2.686 | 2.716 | 2.743 | 2.767 |
Two-sided test, α = 0.01
| Treatment groups compared with control (k) → df ↓ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 4.032 | 4.627 | 4.975 | 5.219 | 5.406 | 5.557 | 5.683 | 5.792 | 5.887 | 5.971 | 6.047 | 6.116 |
| 6 | 3.707 | 4.212 | 4.506 | 4.711 | 4.869 | 4.997 | 5.104 | 5.196 | 5.276 | 5.347 | 5.411 | 5.469 |
| 7 | 3.499 | 3.948 | 4.208 | 4.389 | 4.529 | 4.642 | 4.736 | 4.817 | 4.888 | 4.951 | 5.008 | 5.059 |
| 8 | 3.355 | 3.766 | 4.002 | 4.168 | 4.295 | 4.397 | 4.483 | 4.557 | 4.621 | 4.679 | 4.730 | 4.777 |
| 9 | 3.250 | 3.633 | 3.853 | 4.006 | 4.124 | 4.219 | 4.299 | 4.367 | 4.427 | 4.480 | 4.528 | 4.571 |
| 10 | 3.169 | 3.531 | 3.739 | 3.883 | 3.994 | 4.084 | 4.159 | 4.223 | 4.279 | 4.329 | 4.374 | 4.415 |
| 11 | 3.106 | 3.452 | 3.649 | 3.787 | 3.892 | 3.978 | 4.049 | 4.110 | 4.164 | 4.211 | 4.254 | 4.293 |
| 12 | 3.055 | 3.387 | 3.577 | 3.709 | 3.811 | 3.892 | 3.960 | 4.019 | 4.070 | 4.116 | 4.157 | 4.194 |
| 13 | 3.012 | 3.335 | 3.518 | 3.646 | 3.743 | 3.822 | 3.888 | 3.944 | 3.994 | 4.038 | 4.077 | 4.113 |
| 14 | 2.977 | 3.290 | 3.468 | 3.592 | 3.687 | 3.763 | 3.827 | 3.882 | 3.930 | 3.972 | 4.011 | 4.045 |
| 15 | 2.947 | 3.253 | 3.426 | 3.547 | 3.639 | 3.713 | 3.776 | 3.829 | 3.875 | 3.917 | 3.954 | 3.988 |
| 16 | 2.921 | 3.220 | 3.390 | 3.508 | 3.598 | 3.671 | 3.731 | 3.783 | 3.829 | 3.869 | 3.905 | 3.938 |
| 17 | 2.898 | 3.192 | 3.359 | 3.474 | 3.563 | 3.634 | 3.693 | 3.744 | 3.788 | 3.828 | 3.863 | 3.896 |
| 18 | 2.878 | 3.168 | 3.331 | 3.445 | 3.531 | 3.601 | 3.659 | 3.709 | 3.753 | 3.792 | 3.826 | 3.858 |
| 19 | 2.861 | 3.146 | 3.307 | 3.419 | 3.504 | 3.572 | 3.630 | 3.679 | 3.722 | 3.760 | 3.794 | 3.825 |
| 20 | 2.845 | 3.127 | 3.285 | 3.395 | 3.479 | 3.547 | 3.603 | 3.651 | 3.694 | 3.731 | 3.765 | 3.795 |
| 24 | 2.797 | 3.067 | 3.218 | 3.323 | 3.403 | 3.468 | 3.521 | 3.567 | 3.608 | 3.643 | 3.675 | 3.704 |
| 30 | 2.750 | 3.009 | 3.154 | 3.254 | 3.330 | 3.391 | 3.442 | 3.486 | 3.524 | 3.558 | 3.589 | 3.616 |
| 40 | 2.704 | 2.952 | 3.091 | 3.186 | 3.259 | 3.317 | 3.366 | 3.408 | 3.444 | 3.476 | 3.505 | 3.531 |
| 60 | 2.660 | 2.898 | 3.030 | 3.121 | 3.190 | 3.246 | 3.292 | 3.332 | 3.366 | 3.397 | 3.424 | 3.449 |
| 120 | 2.617 | 2.845 | 2.972 | 3.059 | 3.124 | 3.177 | 3.221 | 3.259 | 3.291 | 3.320 | 3.346 | 3.370 |
| ∞ | 2.576 | 2.794 | 2.915 | 2.998 | 3.060 | 3.110 | 3.152 | 3.188 | 3.219 | 3.246 | 3.271 | 3.293 |
One-sided test, α = 0.05
| Treatment groups compared with control (k) → df ↓ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 2.015 | 2.440 | 2.681 | 2.848 | 2.976 | 3.078 | 3.163 | 3.236 | 3.300 | 3.356 | 3.407 | 3.453 |
| 6 | 1.943 | 2.337 | 2.558 | 2.711 | 2.827 | 2.920 | 2.998 | 3.064 | 3.122 | 3.174 | 3.220 | 3.261 |
| 7 | 1.895 | 2.267 | 2.476 | 2.619 | 2.728 | 2.815 | 2.888 | 2.950 | 3.004 | 3.052 | 3.095 | 3.134 |
| 8 | 1.860 | 2.217 | 2.416 | 2.553 | 2.657 | 2.740 | 2.809 | 2.868 | 2.919 | 2.965 | 3.006 | 3.043 |
| 9 | 1.833 | 2.180 | 2.372 | 2.504 | 2.604 | 2.684 | 2.750 | 2.807 | 2.856 | 2.900 | 2.939 | 2.974 |
| 10 | 1.812 | 2.151 | 2.338 | 2.466 | 2.562 | 2.640 | 2.704 | 2.759 | 2.807 | 2.849 | 2.887 | 2.921 |
| 11 | 1.796 | 2.127 | 2.310 | 2.435 | 2.529 | 2.605 | 2.667 | 2.721 | 2.768 | 2.809 | 2.846 | 2.879 |
| 12 | 1.782 | 2.108 | 2.287 | 2.410 | 2.502 | 2.576 | 2.638 | 2.690 | 2.735 | 2.776 | 2.812 | 2.845 |
| 13 | 1.771 | 2.092 | 2.269 | 2.389 | 2.480 | 2.552 | 2.613 | 2.664 | 2.709 | 2.748 | 2.784 | 2.816 |
| 14 | 1.761 | 2.079 | 2.253 | 2.371 | 2.461 | 2.532 | 2.592 | 2.642 | 2.686 | 2.725 | 2.760 | 2.791 |
| 15 | 1.753 | 2.067 | 2.239 | 2.356 | 2.444 | 2.515 | 2.573 | 2.623 | 2.667 | 2.705 | 2.740 | 2.771 |
| 16 | 1.746 | 2.057 | 2.227 | 2.343 | 2.430 | 2.500 | 2.558 | 2.607 | 2.650 | 2.688 | 2.722 | 2.753 |
| 17 | 1.740 | 2.048 | 2.217 | 2.332 | 2.418 | 2.487 | 2.544 | 2.593 | 2.635 | 2.673 | 2.706 | 2.737 |
| 18 | 1.734 | 2.040 | 2.208 | 2.321 | 2.407 | 2.475 | 2.532 | 2.580 | 2.622 | 2.660 | 2.693 | 2.723 |
| 19 | 1.729 | 2.033 | 2.200 | 2.312 | 2.397 | 2.465 | 2.521 | 2.569 | 2.611 | 2.648 | 2.681 | 2.711 |
| 20 | 1.725 | 2.027 | 2.192 | 2.304 | 2.389 | 2.456 | 2.512 | 2.559 | 2.601 | 2.637 | 2.670 | 2.699 |
| 24 | 1.711 | 2.008 | 2.170 | 2.279 | 2.362 | 2.427 | 2.482 | 2.528 | 2.569 | 2.604 | 2.636 | 2.665 |
| 30 | 1.697 | 1.989 | 2.147 | 2.255 | 2.335 | 2.399 | 2.453 | 2.498 | 2.537 | 2.572 | 2.603 | 2.631 |
| 40 | 1.684 | 1.970 | 2.125 | 2.230 | 2.309 | 2.372 | 2.424 | 2.468 | 2.506 | 2.540 | 2.570 | 2.598 |
| 60 | 1.671 | 1.952 | 2.104 | 2.207 | 2.284 | 2.345 | 2.395 | 2.439 | 2.476 | 2.509 | 2.538 | 2.565 |
| 120 | 1.658 | 1.934 | 2.083 | 2.183 | 2.258 | 2.318 | 2.368 | 2.410 | 2.446 | 2.478 | 2.507 | 2.533 |
| ∞ | 1.645 | 1.916 | 2.062 | 2.160 | 2.234 | 2.292 | 2.340 | 2.381 | 2.417 | 2.448 | 2.476 | 2.502 |
One-sided test, α = 0.01
| Treatment groups compared with control (k) → df ↓ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 3.365 | 3.900 | 4.211 | 4.429 | 4.597 | 4.733 | 4.846 | 4.944 | 5.030 | 5.106 | 5.174 | 5.236 |
| 6 | 3.143 | 3.607 | 3.876 | 4.064 | 4.208 | 4.324 | 4.422 | 4.505 | 4.579 | 4.644 | 4.702 | 4.755 |
| 7 | 2.998 | 3.418 | 3.660 | 3.828 | 3.957 | 4.062 | 4.149 | 4.224 | 4.290 | 4.348 | 4.400 | 4.448 |
| 8 | 2.896 | 3.286 | 3.509 | 3.665 | 3.784 | 3.880 | 3.960 | 4.029 | 4.089 | 4.143 | 4.191 | 4.234 |
| 9 | 2.821 | 3.189 | 3.399 | 3.545 | 3.656 | 3.746 | 3.821 | 3.886 | 3.942 | 3.992 | 4.037 | 4.078 |
| 10 | 2.764 | 3.115 | 3.314 | 3.453 | 3.559 | 3.644 | 3.715 | 3.777 | 3.830 | 3.878 | 3.920 | 3.959 |
| 11 | 2.718 | 3.056 | 3.247 | 3.380 | 3.482 | 3.564 | 3.632 | 3.690 | 3.742 | 3.787 | 3.828 | 3.865 |
| 12 | 2.681 | 3.008 | 3.193 | 3.322 | 3.420 | 3.499 | 3.564 | 3.621 | 3.670 | 3.714 | 3.753 | 3.789 |
| 13 | 2.650 | 2.969 | 3.149 | 3.274 | 3.368 | 3.445 | 3.509 | 3.563 | 3.611 | 3.654 | 3.692 | 3.726 |
| 14 | 2.624 | 2.936 | 3.111 | 3.233 | 3.325 | 3.400 | 3.462 | 3.515 | 3.562 | 3.603 | 3.640 | 3.674 |
| 15 | 2.602 | 2.908 | 3.080 | 3.198 | 3.289 | 3.362 | 3.422 | 3.474 | 3.520 | 3.560 | 3.596 | 3.629 |
| 16 | 2.583 | 2.884 | 3.052 | 3.169 | 3.257 | 3.329 | 3.388 | 3.439 | 3.484 | 3.523 | 3.559 | 3.591 |
| 17 | 2.567 | 2.862 | 3.028 | 3.143 | 3.230 | 3.300 | 3.358 | 3.409 | 3.452 | 3.491 | 3.526 | 3.557 |
| 18 | 2.552 | 2.844 | 3.007 | 3.120 | 3.206 | 3.275 | 3.332 | 3.382 | 3.425 | 3.463 | 3.497 | 3.528 |
| 19 | 2.539 | 2.827 | 2.989 | 3.100 | 3.185 | 3.253 | 3.309 | 3.358 | 3.400 | 3.438 | 3.472 | 3.502 |
| 20 | 2.528 | 2.813 | 2.972 | 3.082 | 3.166 | 3.233 | 3.289 | 3.337 | 3.378 | 3.416 | 3.449 | 3.479 |
| 24 | 2.492 | 2.767 | 2.921 | 3.027 | 3.107 | 3.171 | 3.225 | 3.271 | 3.311 | 3.346 | 3.378 | 3.407 |
| 30 | 2.457 | 2.723 | 2.871 | 2.973 | 3.050 | 3.111 | 3.163 | 3.207 | 3.245 | 3.279 | 3.310 | 3.338 |
| 40 | 2.423 | 2.680 | 2.822 | 2.920 | 2.994 | 3.053 | 3.103 | 3.145 | 3.182 | 3.214 | 3.244 | 3.270 |
| 60 | 2.390 | 2.638 | 2.775 | 2.869 | 2.940 | 2.997 | 3.044 | 3.085 | 3.120 | 3.151 | 3.179 | 3.205 |
| 120 | 2.358 | 2.597 | 2.729 | 2.820 | 2.888 | 2.942 | 2.988 | 3.026 | 3.060 | 3.090 | 3.117 | 3.141 |
| ∞ | 2.326 | 2.558 | 2.685 | 2.772 | 2.837 | 2.889 | 2.933 | 2.970 | 3.002 | 3.031 | 3.056 | 3.079 |
Tip: click any value to highlight its row and column.
How to read this table
- Count k, the number of treatment groups compared with the control. The control itself is not counted.
- Find the error degrees of freedom from the ANOVA: N − (k + 1) for a one-way design with N subjects in total.
- For each treatment, compute t = (x̄ᵢ − x̄₀) / √(MSE (1/nᵢ + 1/n₀)).
- A treatment differs from control if |t| (two-sided) or t (one-sided) exceeds the table value.
Worked example
A dose-ranging study randomizes 7 patients to each of placebo and three doses (N = 28), so k = 3 and error df = 28 − 4 = 24. Row 24, column k = 3 of the two-sided α = 0.05 table gives 2.507. With t-statistics of 1.85, 2.71 and 3.40 for the low, middle and high doses, the middle and high doses differ significantly from placebo, with the familywise error rate held at 5% across all three comparisons.
For comparison, an unadjusted t-test would use t = 2.064 on 24 df, and a Bonferroni correction for three comparisons would use 2.574. Dunnett's value is smaller than Bonferroni's because it accounts for the correlation created by the shared control group.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Critical values are quantiles of the maximum of k equicorrelated t-statistics (correlation 0.5, which applies when all groups are the same size), computed by numerical integration over the multivariate t distribution and rounded to three decimals. The k = 1 column equals Student's t.
Frequently asked questions
What if the group sizes are unequal?
The correlation between comparisons is then not exactly 0.5 and the table is approximate. The approximation is good when the control group is not much larger than the others; for other designs, use software that computes the exact multivariate t probability.
Should the control group be larger?
Often, yes. Dunnett showed that allocating about √k times as many subjects to the control as to each treatment maximizes power for a fixed total sample size.
Dunnett or Tukey?
Use Dunnett when only comparisons with the control matter. It has more power for those comparisons than Tukey's HSD, which also tests every pair of treatments.