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Winning the Lottery

DICE 1
DICE 2
How much information is enough?
Comment

' There is much luck in the world, but it is luck. We are none of us safe '.

So said EM Forster nearly 100 years ago. Bandolier is constantly astonished in its travels by people who appreciate the importance of chance, in, say, winning a lottery or avoiding a car accident, but not in clinical trials. Perhaps it is all down to the way statistics are taught. We should forget probabilities and p-values, and acquaint ourselves with more relevant information, notably how much data do we need to be sure that an observation is not likely to occur just by chance.

Why are people impressed with p-values? The cherished value of 0.05 merely says that a result is not more likely to have occurred by chance than 1 time in 20. Most of us have played Monopoly or other games involved with throwing dice. We will have experienced that throwing two sixes with two dice happens relatively often, yet the chance of that is about 1 time in 36.

Look at it another way. If you were about to cross a bridge, and were told that there was a 1 in 20 chance of it falling down when you were on it, would you take the chance? What about 1 in 100, or 1 in 1000? That p-value of 0.05 also tells you that 1 time in 20 the bridge will fall down.

The dice analogy is pertinent, because there are now (at least) two papers that look at random chance and clinical trials, reminding us how often and how much chance can affect results. An older study actually used dice to mimic clinical trials in stroke prevention [1], while a more recent study [2] used computer simulations of cancer therapy.

DICE 1

In this study [1] participants in a practical class on statistics at a stroke course were given dice and asked to roll them a specified number of times to represent the treatment group of a randomised trial. If six was thrown, this was recorded as a death, with any other number a survival. The procedure was repeated for a control group of similar size. Group size ranged from 5 to 100 patients.

The paper gives the results of all 44 trials for 2,256 'patients'. While the paper does many clever things, it is perhaps more instructive to look at the results of the 44 trials. Since each arm of the trial looks for the throwing of one out of six possibilities for standard dice, we might expect that the rate of events was 16.7% (100/6) in each, with an odds ratio or relative risk of 1.

Figure 1 shows a L'Abbé plot of the 44 trials. The expected result is a grouping in the bottom left, on the line of equality at about 17%. Actually, it is a bit more dispersed than that, with some trials far from the line of equality.

Figure 1: L'Abbé plot of DICE 1 trials

The odds ratios for individual trials are shown in Figure 2. Two trials (20 and 40 in total) had odds ratios statistically different from 1. That's one time in every 22 trials, what we expect by chance.

Figure 2: Odds ratios for individual DICE studies, by number in 'trial'

The variability in individual trial arms is shown in Figure 3, where the results are shown for all 88 trial arms. The vertical line shows the overall result (16.7%). Larger samples come close to this, but small samples show values as low as zero, and as high as 60%.

Figure 3: Percentage of events in each trial arm of DICE 'trials'

The overall result, pooling data from all 44 trials, showed that events occurred in 16.0% of treatments and 17.6% of controls (overall mean 16.7%). The relative risk was 0.8 (0.5 to 1.1) and the NNT was 63, with a 95% confidence interval than went from one benefit for every 21 treatments to one harm for every 67 treatments (Table 1).

Table 1: Meta-analysis of DICE trials, with sensitivity analysis by size of trial

NNT (95% CI)

All trials 44 2256 16.0 17.6 0.8 (0.5 to 1.1) 62 (21 to -67)
Larger trials (>40 per group) 11 1190 19.5 17.8 1.1 (0.9 to 1.4) -60 (36 to -16)
Smaller trials (<40 per group) 33 1066 12.0 17.3 0.7 (0.53 to 0.94) 19 (11 to 98)

Many of the experimental DICE trials were quite small, with as few as five per group. The smaller trials, with 40 per group or less, actually came up with a statistically significant result (Table 1). The NNT here was 19 (11 to 98).

DICE 2

Information on the time between randomisation and death in a control group of 580 patients in a colorectal cancer trial was used to simulate 100 theoretical clinical trials. Each time the same 580 patients were randomly allocated to a theoretical treatment or control group, and survival curves calculated [2].

Four of the trials artificially generated had statistically significant results. One was significant at the 0.003 level (1 in 333) and showed a large theoretical decrease in mortality of 40%.

Subgroup analysis was done for this trial by randomly allocating patients to type A or type B, and doing this 100 times. Over half (55%) of the subgroup analyses showed statistical significance between subgroups. The extremes of results were no difference between subgroups, to a result with high significance of 0.00005 (1 in 20,000). In another trial that had bare statistical significance, four of 100 simulated subgroups had statistical significance at the 1 in 100 level.

How much information is enough?

While it is relatively easy to demonstrate that inadequate amounts of information can result in erroneous conclusions, the alternative question, how much information we need to avoid erroneous conclusions, is more difficult to answer. It depends on a number of things. Two important issues are the size of the effect you are looking at (absolute differences between treatment and control), and how sure you want to be.

A worked example using simulations of acute pain trials [3] gives us some idea. Using the same 16% event rate as in DICE 1 as the rate with controls (because it happens to be what is found with placebo), it looked at event rates with treatment of 40%, 50% and 60%, equivalent to NNTs of 4.2, 2.9 and 2.3. The numbers in treatment and placebo group were each simulated from 25 patients per group (trial size 50) to 500 patients per group (trial size 1000). For each condition 10,000 trials were simulated and the percentage where the NNT was within ±0.5 of the true NNT counted.

The results are shown in Table 2. With 1000 patients in a trial where the NNT was 2.3, we could be 100% sure that the NNT measured was within ±0.5 of the true NNT; all trials of this size would produce values between 1.8 and 2.8. In a trial of 50 patients where the NNT was 4.2, only one in four trials would produce an NNT within ±0.5; the true value is between 3.7 and 4.7, and three-quarters of trials (or meta-analyses) of this size would produce NNTs below 3.7 or over 4.7.

Table 2: Effect of size and size of effect on confidence of treatment effect

Group size

25 26 37 57
50 28 51 73
100 38 61 88
200 55 81 96
300 63 89 99
400 71 93 99
500 74 95 100
With control the event rate was 16%
At least
50% within +/- 0.5
80% within +/- 0.5
95% within +/- 0.5

The study also shows that to be certain of the size of the effect (the NNT, say), we need ten times more information than just to know that there is statistical significance.

Comment

What does all this tell us? It emphasises that the random play of chance is a factor we cannot ignore, and that small trials are more prone to chance effects than larger ones. And it is not just an effect seen in single trials. Even when we pool data from small trials just from rolling dice, as in DICE 1, a meta-analysis can come up with a statistically significant effect when there was none.

High levels of statistical significance can be generated just by the random play of chance. DICE 2 found levels of statistical significance of 1 in 333 for at least one simulated trial, and 1 in 20,000 for a subgroup analysis of that trial.

Not only do we need well-conducted trials of robust design and reporting, we also need large amounts of information if the size of a clinical effect is to be accurately assessed. The rule of thumb is that where the difference between control and treatment is small we need very large amounts. Only when the difference is large (an absolute risk increase or decrease of 50%, affecting every second patient) can we be reasonably happy with information from 500 patients or fewer.

When we see differences between trials, or between responses to placebo, the rush is often to try and explain the difference according to some facet of trial design or patient characteristic. Almost never does anyone ask how likely the difference is to occur just by the random play of chance.

Some things are very unlikely, like winning the lottery. It's the random play of chance, coupled with low downside (£1) and high upside (£millions and our only hope of early retirement to a gin palace) that makes it worthwhile.

References:

1. CE Counsell et al. The miracle of DICE therapy for acute stroke: fact or fictional product of subgroup analysis? BMJ 1994 309: 1677-1681.
2. M Clarke, J Halsey. DICE2: a further investigation of the effects of chance in life, death and subgroup analyses. International Journal of Clinical Practice 2001 55: 240-242.
3. RA Moore et al. Size is everything - large amounts of information are needed to overcome random effects in estimating direction and magnitude of treatment effects. Pain 1998 78: 209-16.

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