True difference

This solution is of little practical use, because small inaccuracies in aj — a, lead to large errors in large terms with alternating signs, swamping the true differences. Clayton et al. (1961) derived an approximation, based on the Poisson-like distribution obtained when all the cross-sections are equal ... 

The first precise or calculable aspect of experimental design encountered is determining sufficient test and control group sizes to allow one to have an adequate level of confidence in the results of a study (that is, in the ability of the study design with the statistical tests used to detect a true difference, or effect, when it is present). The statistical test contributes a level of power to such a detection. Remember that the power of a statistical test is the probability that a test results in rejection of a hypothesis, H0 say, when some other hypothesis, H, say, is valid. This is termed the power of the test with respect to the (alternative) hypothesis H. ... 

This leads to the term Power (1 - j3), which quantifies the ability of the study to find the true differences of various values of S. It is the probability of rejecting the null hypothesis when it is false or determining that the alternative hypothesis is true when indeed it is true. 

Of particular interest is the gender distribution of the Cluster B personality disorders. Three out of four patients diagnosed with BPD are women. Likewise, most patients diagnosed with HPD are females as well. In contrast, ASPD is much more common among men. There has been considerable discussion as to whether these are true differences or whether they result from diagnostic biases. 

The aim of any clinical trial is to have low risk of Type I and II errors and sufficient power to detect a difference between treatments, if it exists. Of the three factors in determining sample size, the power (probability of detecting a true difference) is arbitrarily chosen. The magnitude of the drug s effect can be estimated with more or less accuracy from previous experience with drugs of the same or similar action, and the variability of the measurements is often known from published experiments on the primary endpoint, with or without the drug. These data will, however, not be available for novel substances in a new class and frequently the sample size in the early phase of development has to be chosen on an arbitrary basis. 

We see that in contrast to the type-1 error, the type-11 error is defined as occurring when accepting the null hypothesis if it is false. The power of a test is defined to be the probability of detecting a true difference and is equal to 1 — probability (type-11 error). The type-11 error and power depend upon the type-1 error, the sample size, the clinically relevant difference (CRD) that we are interested in detecting and the expected variability. Where do these values come from ... 

Capillary zone electrophoresis is a separation technique that benefits from very high efficiency, not selectivity. This is in contrast to chromatography, for which the converse is true. Differences in mobility in the range of 0.01% can be enough for complete resolution of neighboring peaks. The resolution R is defined as... 

Suppose that the value of icj — X2 turns out to be 1.4mmol/l. We know full well that this will not be equal to the true difference in the means, pi — p2-also know that if we were to repeat the trial under identical circumstances, same protocol, same investigator and so on, but of course a different sample of patients, then we would come up with a different value for Xj — X2. 

The interpretation of this interval is essentially as before we can be 95 per cent confident that the true difference in the (population) means, — p.2> is between 0.8 and 2.0. In other words the data are telling us that the mean reduction p. in... 

The quantity — U2 is the vertical distance between the lines and represents the (adjusted) difference in the mean time to recurrence in the test treatment group minus the mean time to recurrence in the control group the treatment effect. It is straightforward also to obtain a confidence interval around this adjusted treatment effect to capture the true difference. 

When we say that a trial has 80 per cent power to detect a certain level of effect, for example 4 mmHg, what we mean is that if we conduct the trial and the true difference really is 4 mmHg then there is an 80 per cent chance of coming out of the trial with a significant p-value, and declaring differences. 

So, for example, if the true difference between the treatments means was 0.50 mmol/1 then this trial would have a 62.3 per cent chance of coming out with... 

The next step is to undertake the trial and calculate the 95 per cent confidence interval for the difference in the means (mean increase in PEF on new inhaler (pi) - mean increase in PEF on existing inhaler ( 2))- As a first example, suppose that this confidence interval is (-71/min, 121/min). In other words, we can be 95 per cent confident that the true difference, pj — P2, is between 71/min in favour of the existing inhaler and 121/min in favour of the new inhaler. 

This method of calculating the conditional power assumes that the observed difference between the treatments at the interim stage is the true difference, termed the conditional power under the current trend. It is also possible to calculate conditional power under other assumptions, for example, that the true treatment difference in the remaining part of the trial following the interim analysis is equal to d. These calculations under different assumptions about how the future data should behave will provide a broad basis on which to make judgements about terminating the trial for futility. 

Is the study valid Did the investigators use a randomized, controlled, blinded design in which all the patients were followed up at the end Apart from intrinsic differences in the treatments themselves, were all the patients treated the same way Without affirming these relatively straightforward parameters, it is impossible to know whether differences in the outcome reflect true differences in the impact of the treatments or some other characteristic of the study. 

A finding of no significant difference in outcome between two treatments is ambiguous unless a non-inferiority design is used. Two-arm non-inferiority designs require much larger sample sizes to test adequately the statistical hypothesis of no true difference in efficacy between a test and reference drug. 

For sample size = 50, to be 95% sure that an observed difference in cancer incidence between experimental and control groups indicates a true difference in incidence that is greater than zero, we must observe an incidence difference greater than 23%. The probability, however, of observing an incidence of less than 23% if the true difference is greater than zero but less than 23% varies from 50% to 95%. For example, if the true difference in incidence is 5.2%, the probability of observing values of less than 23% is about 89%. 

True Difference in Cancer Incidence Between Control and Experimental Groups in Percent... 

Limiting false-positive errors to 5 percent, however, increases the likelihood of false negatives. Assume, for instance, that the true incidence difference in the example shown in Figure 1 is 5.2 percent. To keep false positive errors to less than 5 percent, all differences of less than 23 percent are deemed too small to be the result of exposure. If the true difference in cancer incidence is, in fact, 5.2 percent, the probability of erroneously concluding that a substance is not a carcinogen is more than 89 percent. ... 

Available evidence indicates, then, that ORD-CD studies reflect a real property of membrane protein. However, results from different laboratories vary. The lack of agreement may reflect true differences in protein conformation from membrane to membrane, but in some cases the effects observed might be artifacts arising, for example, from different preparative procedures or from some poorly understood property of scattering systems. 

The empirical tests described below determine the chalking and weathering differences between two pigments, but do not always give the true differences. Other test methods have therefore been developed, e.g., determination of mass losses on weathering (gravimetric test) [1.51]. 

PhCO groups act mainly through their polar effects. The somewhat larger k0 for the PhCO derivative is consistent with the larger cF value of the PhCO group the difference in the log/cD values of 0.35 actually underestimates the true difference because k0 for the PhCO derivative was determined at 20°C rather than 25°C. 

When there are intervariable correlations, another source of error associated with the OVAT approach appears, the so-called type II error. This means that a true difference is spuriously undetected. The Bonferoni adjustment of p-values is one rich source of increased type II errors in univariate analysis of multivariate data. This is easily realized if a situation is considered where the effect of a drug has been recorded on one relevant variable and nine irrelevant variables. The Bonferoni adjustment would in this case obscure the truly significant change in the relevant variable by the compensation for the irrelevant variables. 

If the enclosed impeller case is not constant, then Vr = constantIrB and the pressure variation due to the radial velocity will be somewhat different from that due to rotation alone B is the diameter of the pipe receiving the flow. The true difference in pressure between any two points may then be determined by adding together the pressure differences caused by these two types of flow considered separately. This procedure is sometimes convenient when friction losses are to be considered and where the losses for the two types of flow are considered to vary in different ways. However, for the case where H is consid-... 

Apparent Layer Thickness. Many images of sheet silicates and clay minerals exhibit layers of different thicknesses. These thickness variations commonly are measured from micrographs and interpreted as representing true differences in thickness. However, simulations have shown that imperfect orientation and focus can cause the... 

Figure 6.4 tells us that, if we were to collect more and more data, the difference in clearance would not necessarily remain at the point estimate of +0.298 ml/min/kg that our small samples yielded. Ultimately we might find the true difference is actually as small as +0.142 or as large as +0.455 ml/min/kg. 

Figure 8.2 Power curve for the theophylline clearance experiment (assumes the true difference is an increase of 0.3 ml/min/kg and the SD among clearances is 0.21 ml/min/kg)...

Two experiments may produce exactly the same P value, but that does not mean that they necessarily lead to the same level of certainty that there is a true difference in outcome. 

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