The sample size of each data set was 10 sequences (in nine series) and of phylogenetic relationships of RNA viruses in population and epidemiological studies. .. Exact, unbiased estimates of P values in contingency tables were. In Section , we talked about differences and P-values in a general way. . The increased sample size has allowed us to conclude that the difference between . effect of a large number of different anti-microbial agents on a bacterium and a virus. Thus a simple comparison of 95% confidence intervals cannot be made. Since the size of the sample influences the value of t, the size of the sample is taken . for example if the ratio of the larger to the smaller is greater than two, then.
For example, we may be testing a mutant that we suspect changes the ratio of male-to-hermaphrodite cross-progeny following mating. In this case, the null hypothesis is that the mutant does not differ from wild type, where the sex ratio is established to be 1: More directly, the null hypothesis is that the sex ratio in mutants is 1: Furthermore, the complement of the null hypothesis, known as the experimental or alternative hypothesis, would be that the sex ratio in mutants is different than that in wild type or is something other than 1: For this experiment, showing that the ratio in mutants is significantly different than 1: Whether or not a result that is statistically significant is also biologically significant is another question.
Moreover, the term significant is not an ideal one, but because of long-standing convention, we are stuck with it. Statistically plausible or statistically supported may in fact be better terms. We interpret this to mean that even if there was no actual difference between the mutant and wild type with respect to their sex ratios, we would still expect to see deviations as great, or greater than, a 6: Put another way, if we were to replicate this experiment times, random chance would lead to ratios at least as extreme as 6: Of course, you may well wonder how it is possible to extrapolate from one experiment to make conclusions about what approximately the next 99 experiments will look like.
There is well-established statistical theory behind this extrapolation that is similar in nature to our discussion on the SEM. In any case, a large P-value, such as 0. It is, however, possible that a true difference exists but that our experiment failed to detect it because of a small sample size, for instance. In contrast, suppose we found a sex ratio of 6: In this case, the likelihood that pure chance has conspired to produce a deviation from the 1: Because this is very unlikely, we would conclude that the null hypothesis is not supported and that mutants really do differ in their sex ratio from wild type.
Such a finding would therefore be described as statistically significant on the basis of the associated low P-value. Of course, common sense would dictate that there is no rational reason for anointing any specific number as a universal cutoff, below or above which results must either be celebrated or condemned. Can anyone imagine a convincing argument by someone stating that they will believe a finding if the P-value is 0. Even a P-value of 0. Well, for one thing, it makes life simpler for reviewers and readers who may not want to agonize over personal judgments regarding every P-value in every experiment.
It could also be argued that, much like speed limits, there needs to be an agreed-upon cutoff. Even if driving at 76 mph isn't much more dangerous than driving at 75 mph, one does have to consider public safety.
In the case of science, the apparent danger is that too many false-positive findings may enter the literature and become dogma. Noting that the imposition of a reasonable, if arbitrary, cutoff is likely to do little to prevent the publication of dubious findings is probably irrelevant at this point. The key is not to change the chosen cutoff—we have no better suggestion 12 than 0. The key is for readers to understand that there is nothing special about 0.
Judgment and common sense should always take precedent over an arbitrary number. However, most texts don't bother and so we won't either. Most but not quite all of the values will span a range of approximately four SDs. However, you can see that the distribution of the sample won't necessarily be perfectly symmetric and bell-shape, though it is close.
Also note that just because the distribution in Panel A is bimodal does not imply that classical statistical methods are inapplicable.
In fact, a simulation study based on those data showed that the distribution of the sample mean was indeed very close to normal, so a usual t-based confidence interval or test would be valid. This is so because of the large sample size and is a predictable consequence of the Central Limit Theorem see Section 2 for a more detailed discussion. Changing the sample design e. Had we used a much larger number of trials e.
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Fisher, a giant in the field of statistics, chose this value as being meaningful for the agricultural experiments with which he worked in the s. Comparing two means 2. Introduction Many studies in our field boil down to generating means and comparing them to each other.
This is true even if the data are acquired from a single population; the sample means will always be different from each other, even if only slightly.
The pertinent question that statistics can address is whether or not the differences we inevitably observe reflect a real difference in the populations from which the samples were acquired. Put another way, are the differences detected by our experiments, which are necessarily based on a limited sample size, likely or not to result from chance effects of sampling i. If chance sampling can account for the observed differences, then our results will not be deemed statistically significant In contrast, if the observed differences are unlikely to have occurred by chance, then our results may be considered significant in so much as statistics are concerned.
Whether or not such differences are biologically significant is a separate question reserved for the judgment of biologists. Most biologists, even those leery of statistics, are generally aware that the venerable t-test a. Several factors influence the power of the t-test to detect significant differences. These include the size of the sample and the amount of variation present within the sample.
If these sound familiar, they should. They were both factors that influence the size of the SEM, discussed in the preceding section. This is not a coincidence, as the heart of a t-test resides in estimating the standard error of the difference between two means SEDM.
Greater variance in the sample data increases the size of the SEDM, whereas higher sample sizes reduce it.
Thus, lower variance and larger samples make it easier to detect differences. If the size of the SEDM is small relative to the absolute difference in means, then the finding will likely hold up as being statistically significant.
In fact, it is not necessary to deal directly with the SEDM to be perfectly proficient at interpreting results from a t-test. We will therefore focus primarily on aspects of the t-test that are most relevant to experimentalists. These include choices of carrying out tests that are either one- or two-tailed and are either paired or unpaired, assumptions of equal variance or not, and issues related to sample sizes and normality. We would also note, in passing, that alternatives to the t-test do exist.
These tests, which include the computationally intensive bootstrap see Section 6. For reasonably large sample sizes, a t-test will provide virtually the same answer and is currently more straightforward to carry out using available software and websites. It is also the method most familiar to reviewers, who may be skeptical of approaches that are less commonly used. We will do this through an example.
Imagine that we are interested in knowing whether or not the expression of gene a is altered in comma-stage embryos when gene b has been inactivated by a mutation. To look for an effect, we take total fluorescence intensity measurements 15 of an integrated a:: For each condition, we analyze 55 embryos. Expression of gene a appears to be greater in the control setting; the difference between the two sample means is Summary of GFP-reporter expression data for a control and a test group.
Along with the familiar mean and SD, Figure 5 shows some additional information about the two data sets. Recall that in Section 1. What we didn't mention is that distribution of the data 16 can have a strong impact, at least indirectly, on whether or not a given statistical test will be valid.
Such is the case for the t-test.Statistics 101: Confidence Intervals, Estimating Sample Size Needed
Looking at Figure 5we can see that the datasets are in fact a bit lopsided, having somewhat longer tails on the right. In technical terms, these distributions would be categorized as skewed right. Although not critical to our present discussion, several parameters are typically used to quantify the shape of the data including the extent to which the data deviate from normality e.
In any case, an obvious question now becomes, how can you know whether your data are distributed normally or at least normally enoughto run a t-test? Before addressing this question, we must first grapple with a bit of statistical theory. The Gaussian curve shown in Figure 6A represents a theoretical distribution of differences between sample means for our experiment.
Put another way, this is the distribution of differences that we would expect to obtain if we were to repeat our experiment an infinite number of times. Thus, if we carried out such sampling repetitions with our two populations ad infinitum, the bell-shaped distribution of differences between the two means would be generated Figure 6A.
Note that this theoretical distribution of differences is based on our actual sample means and SDs, as well as on the assumption that our original data sets were derived from populations that are normal, which is something we already know isn't true. There are several ways we can obtain greater statistical power.
One way is to increase the size of the effect by increasing the size of the experimental factor. An example would be to try to produce a larger effect in a drug trial by increasing the dosage of the drug. Another way is to reduce the amount of uncontrolled variation in the results. For example, standardizing your method of data collection, reducing the number of different observers conducting the experiment, using less variable experimental subjects, and controlling the environment of the experiment as much as possible are all ways to reduce uncontrolled variability.
A third way of increasing statistical power is to change the design of the experiment in a way that allows you to conduct a more powerful test. For example, having equal numbers of replicates in all of your treatments usually increases the power of the test. Simplifying the design of the experiment may increase the power of the test.
Using a more appropriate test can also increase statistical power. Finally, increasing the sample size or number of replicates nearly always increases the statistical power. Obviously the practical economics of time and money place a limit on the number of replicates you can have.
Theoretically, the outcome of an experiment should be equally interesting regardless of whether the outcome of an experiment shows a factor to have a significant effect or not.
As a practical matter, however, there are far more experiments published showing significant differences than studies showing factors to not be significant. There is an important practical reason for this. If an experiment shows differences that are significant, then we assume that is because the factor has a real effect. However, if an experiment fails to show significant differences, this could be because the factor doesn't really have any effect.
But it could also be that the factor has an effect but the experiment just didn't have enough statistical power to detect it. The latter possibility has less to do with the biology involved and more to do with the experimenter's possible failure at planning and experimental design - not something that a scientific journal is going to want to publish a paper about! Generally, in order to publish experiments that do not have significant differences it is necessary to conduct a power test.
A power test is used to show that a test would have been capable of detecting differences of a certain size if those differences had existed. This has already been done on the lab computers but if you are using a computer elsewhere, you may need to enable it. Click on Data Analysis in the Analysis section.
Select Descriptive Statistics, then click OK. Click on the Input Range selection button, then select the range of cells for the column. If there is a label for the column, click the "Labels in first row" checkbox and include it when you select the range of cells. Check the Confidence Level for Mean: You may also wish to check the Summary statistics checkbox as well if you want to calculate the mean value.
To put the results on the same sheet as the column of numbers, click on the Output Range radio button then click on the selection button. Click on the upper left cell of the area of the sheet where you would like for the results to go.
They applied Serratia marcescens bacteria to the hands of their test subjects and measured the number of bacteria that they could extract from the hands with and without washing with the antimicrobial agent. Taking the logarithm of the counts changes the shape of the distribution to one closer to the normal curve see Leyden et al. Thus we see this kind of data transformation being used in all three of the papers. By examining this table, we can see whether any two cleansing agents produced significantly different declines.
For example, since our experiment will be focused on consumer soap containing triclosan, we would like to know if Sickbert-Bennett et al.
After the first episode of handwashing, the mean log reduction of the triclosan soap was greater 1. An obvious weakness is that the test does not produce a numeric measure of the degree of significance. It simply indicates whether P is more or less than 0.
Another is that it can be a more conservative test than necessary. However, confidence intervals can actually overlap by a small amount and the difference still be significant.
Another more subtle problem occurs when more than two groups are being compared. The more groups that are being compared, the more possible pairwise comparisons there are that can be made between groups. This increase in possible pairwise comparisons does not increase linearly with the number of groups. If the alpha level for each comparison is left at 0.
In a scathing response, Paulson points out that Sickbert-Bennett et al.
A biologist's guide to statistical thinking and analysis
According to Paulson, this mistake Point 3 in his paper along with several other statistical and procedural errors made their conclusions meaningless. This paper, along with the response of Kampf and Kramer make interesting reading as they show how a paper can be publically excoriated for poor experimental design and statistical analysis.
Efficacy of hand hygiene agents at short application times.