Why is bias affected when a clinical trial is terminated at an early stage? An interim analysis is an analysis of the data at one or more time points prior the official close of the study with the intention of, e.g., possibly terminating the study early.
According to Piantadosi, S. (Clinical trials - a methodologic perspective):
"The estimate of a treatment effect will be biased when a trial is terminated at an early stage. The earlier the decision, the larger the bias."
Can you explain me this claim. I can easily understand that the accuracy is going to be affected, but the claim about the bias is not obvious to me...
 A: Here is an illustration of how bias might arise in conclusions, and why it may not be the full story.  Suppose you have a sequential trial of a drug which is expected to have a positive (+1) effect but may have a negative effect (-1).  Five guinea pigs are tested one after the other.  The unknown probability of a positive outcome in a single case is in fact $\frac{3}{4}$ and a negative outcome $\frac{1}{4}$. 
So after five trials the probabilities of the different outcomes are 
 Outcome     Probability
+5-0 = +5    243/1024
+4-1 = +3    405/1024
+3-2 = +1    270/1024
+2-3 = -1     90/1024
+1-4 = -3     15/1024
+0-5 = -5      1/1024

so the probability of a positive outcome overall is 918/1024 = 0.896, and the mean outcome is +2.5. Dividing by the 5 trials, this is an average of a +0.5 outcome per trial. 
It is the unbiased figure, as it is also $+1\times\frac{3}{4}-1\times\frac{1}{4}$.
Suppose that in order to protect guinea pigs, the study will be terminated if at any stage the cumulative outcome is negative. Then the probabilities become 
 Outcome     Probability
+5-0 = +5    243/1024
+4-1 = +3    324/1024
+3-2 = +1    135/1024
+2-3 = -1     18/1024
+1-2 = -1     48/1024
+0-1 = -1    256/1024

so the probability of a positive outcome overall is 702/1024 = 0.6855, and the mean outcome is +1.953.  If we looked the mean value of outcome per trial in the previous calculation, i.e. using $\frac{+5}{5}$, $\frac{+3}{5}$, $\frac{+1}{5}$, $\frac{-1}{5}$, $\frac{-1}{3}$ and $\frac{-1}{1}$ then we would get +0.184. 
These are the senses in which there is bias by stopping early in the second scheme, and the bias is in the predicted direction.  But it is not the full story.
Why do whuber and probabilityislogic think stopping early should produce unbiased results?  We know the expected outcome of the trials in the second scheme is +1.953.  The expected number of trials turns out to be 3.906.  So dividing one by the other we get +0.5, exactly as before and what was described as unbiased.
A: First of all, you have to note the context: this only applies when the trial was stopped early due to interim monitoring showing efficacy/futility, not for some random outside reason. In that case the estimate of the effect size will be biased in a completely statististical sense. If you stopped for efficacy, the estimated effect will be too high (assuming it is positive), if you stopped for futility, it will be too low. 
Piantodosi does give an intuitive explanantion as well (Sec 10.5.4 in my edition). Suppose the true difference in two means is 1 unit. When you run a lot of trials, and look at them at your interim analysis time, some of them will have observed effect sizes much above 1, some much below one, and most around 1 - the distribution will be wide, but symmetric. The estimated effect size at this point would not be very accurate, but would be unbiased. However you only stop and report an effect size if the difference is significant (adjusted for multiple testing), that is the estimate is on the high side. In all other cases you keep going and don't report an estimate. That means that conditional on having stopped early, the distribution of the effect size is not symmetric, and its expected value is above the true value of the estimate.
The fact that this effect is more severe early on comes from the larger hurdle for stopping the trial, thus a larger part of the distribution being thrown away during the conditioning.
A: Well, my knowledge on this comes from the Harveian oration in 2008 http://bookshop.rcplondon.ac.uk/details.aspx?e=262
Essentially, to the best of my recollection the results will be biased as 1) stopping early usually means that either the treatment was more or less effective than one hoped, and if this is positive, then you may be capitalising on chance. 
I believe that p values are calculated on the basis of the planned sample size (but i could be wrong on this), and also if you are constantly checking your results to see if any effects have been shown, you need to correct for multiple comparisons in order to insure that you are not merely finding a chance effect.
For example, if you check 20 times for p values below .05 then statistically speaking, you are almost certain to find one significant result.
A: I would disagree with that claim, unless by "bias" Piantadosi means that part of the accuracy which is commonly called bias.  The inference won't be "biased" because you chose to stop per se: it will be "biased" because you have less data.  The so called "likelihood principle" states that inference should only depend on data that was observed, and not on data that might have been observed, but was not.  The LP says
$$P(H|D,S,I)=P(H|D,I)$$
Where $H$ stands for the hypothesis you are testing (in the form of a proposition, such as "the treatment was effective"), $D$ stands for the data you actually observed, and $S$ stands for the proposition "the experiment was stopped early", and $I$ stands for the prior information (such as a model). Now suppose your stopping rule depends on the data $D$ and on the prior information $I$, so you can write $S=g(D,I)$.  Now an elementary rule of logic is $AA=A$ - saying that A is true twice is the same thing as saying it once.
This means that because $S=g(D,I)$ will be true whenever $D$ and $I$ are also true.  So in "boolean algebra" we have $D,S,I = D,g(D,I),I = D,I$.  This proves the above equation of the likelihood principle.  It is only if your stopping rule depends on something other than the data $D$ or the prior information $I$ that it matters.
A: there will be bias (in "statistical sense") if termination of studies is not random. 
In a set of experiments run to conclusion, the "early on" results of (a) some experiments that ultimately find "no effect" will show some effect (as a result of chance) and (b) some experiments that ultimately do find an effect will show "no effect" (likely as a result of lack of power). In a world in which you terminate trials, if you stop (a) more often than (b), you'll end up across run of studies with bias in favor of finding an effect. (Same logic applies for effect sizes; terminating studies that show "bigger than expected" effect early on more often than ones that show "as expected or lower" will inflate count of findings of "big effect.")
If in fact medical trials are terminated when early results show a positive effect -- in order to make treatment available to subjects in placebo or others -- but not when early results are inconclusive, then there will be more type 1 error in such testing than there would be if all experiments were run to conclusion. But that doesn't meant the practice is wrong; the cost of type 1 error, morally speaking, might be lower than denying treatment as quickly as one otherwise would for treatments that really would be shown to work at end of full trial. 
