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This means that the null hypothesis would be written as

That is the test statistic based on the original datathat is being compared to the critical value of the null distribution.

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the distribution under the null hypothesis).

They can be used to test the null hypothesis that a given time series is a martingale process, against the alternative hypothesis that it is a stationary ergodic nonmartingale process.

: This package is for Monte Carlo hypothesis tests, that is,tests using some form of resampling.

Although the above narrative may be ridiculous (indeed, it is meant to be so), the underlying issues are very real. Conclusions based on single -tests, which are not supported by additional complementary data, may well be incorrect. Thus, where does one draw the line? One answer is that no line should be drawn, even in situations where multiple comparison adjustments would seem to be warranted. Results can be presented with corresponding -values, and readers can be allowed to make their own judgments regarding their validity. For larger data sets, such as those from microarray studies, an estimation of either the number or proportion of likely false positives can be provided to give readers a feeling for the scope of the problem. Even without this, readers could in theory look at the number of comparisons made, the chosen significance threshold, and the number of positive hits to come up with a general idea about the proportion of false positives. Although many reviewers and readers may not be satisfied with this kind of approach, know that there are professional statisticians who support this strategy. Perhaps most importantly, understand that whatever approaches are used, data sets, particularly large ones, will undoubtedly contain errors, including both false positives and false negatives. Wherever possible, seek to confirm your own results using multiple independent methods so that you are less likely to be fooled by chance occurrence.

Resampling under the null hypothesis Selecting a null model ..

: This package produces approximately unbiased hypothesistests via bootstrapping.

We show that the tests are consistent against stationary ergodic non-martingale alternatives, and further investigate the nite sample properties of the test statistics through simulation.

The idea:Permutation tests are restricted to the case where the null hypothesis really is null -- that is, that there is no effect.

testing a null hypothesis, bootstrapping is generally ..

This paper proposes general speci cation tests for the martingale hypothesis.

And the -value will answer the question: If the null hypothesis is true, what is the probability that the following result could have occurred by chance sampling?

Of course, our experimental result that GFPwt was greater than GFPmut clearly fails to support this research hypothesis. In such cases, there would be no reason to proceed further with a -test, as the -value in such situations is guaranteed to be >0.5. Nevertheless, for the sake of completeness, we can write out the null hypothesis as

There is nothing to prevent a literal test of Baron and Kenny's statistic. We could
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  • Hypothesis testing using bootstrap resampling - …

    Tests based on these statistics are inconsistent since they just test necessary conditions of the null hypothesis.

  • Hypothesis testing using bootstrap resampling – ..

    Here meaning by a statistical test where the P-value cutoff or “alpha level” (α) is 0.05.

  • the observations that is valid if the null hypothesis is ..

    This discussion assumes that the null hypothesis (of no difference) is true in all cases.

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random variation in a test statistic when there is ..

Along with the familiar mean and SD, shows some additional information about the two data sets. Recall that in , we described what a data set looks like that is normally distributed (). What we didn't mention is that distribution of the data 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 -test. Looking at , we 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 right. Furthermore, because our sample size was sufficiently large (n=55), we can conclude that the populations from whence the data came are also 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.g., , and ). In any case, an obvious question now becomes, how can you know whether your data are distributed normally (or at least normally enough), to run a -test?

cannot reject the null hypothesis that the three ..

To aid in understanding the logic behind the -test, as well as the basic requirements for the -test to be valid, we need to introduce a few more statistical concepts. We will do this through an example. Imagine that we are interested in knowing whether or not the expression of gene is altered in comma-stage embryos when gene has been inactivated by a mutation. To look for an effect, we take total fluorescence intensity measurements of an integrated ::GFP reporter in comma-stage embryos in both wild-type (Control, ) and mutant (Test, ) strains. For each condition, we analyze 55 embryos. Expression of gene appears to be greater in the control setting; the difference between the two sample means is 11.3 billion fluorescence units (henceforth simply referred to as “11.3 units”).

% null hypothesis is that the parameter=mu

The Central Limit Theorem having come to our rescue, we can now set aside the caveat that the populations shown in are non-normal and proceed with our analysis. From we can see that the center of the theoretical distribution (black line) is 11.29, which is the actual difference we observed in our experiment. Furthermore, we can see that on either side of this center point, there is a decreasing likelihood that substantially higher or lower values will be observed. The vertical blue lines show the positions of one and two SDs from the apex of the curve, which in this case could also be referred to as SEDMs. As with other SDs, roughly 95% of the area under the curve is contained within two SDs. This means that in 95 out of 100 experiments, we would expect to obtain differences of means that were between “8.5” and “14.0” fluorescence units. In fact, this statement amounts to a 95% CI for the difference between the means, which is a useful measure and amenable to straightforward interpretation. Moreover, because the 95% CI of the difference in means does not include zero, this implies that the -value for the difference must be less than 0.05 (i.e., that the null hypothesis of no difference in means is not true). Conversely, had the 95% CI included zero, then we would already know that the -value will not support conclusions of a difference based on the conventional cutoff (assuming application of the two-tailed -test; see below).

The Statistical Bootstrap and Other Resampling Methods

The key is to understand that the -test is based on the theoretical distribution shown in , as are many other statistical parameters including 95% CIs of the mean. Thus, for the -test to be valid, the shape of the actual differences in sample means must come reasonably close to approximating a normal curve. But how can we know what this distribution would look like without repeating our experiment hundreds or thousands of times? To address this question, we have generated a complementary distribution shown in . In contrast to , was generated using a computational re-sampling method known as bootstrapping (discussed in ). It shows a histogram of the differences in means obtained by carrying out 1,000 repeats of our experiment. Importantly, because this histogram was generated using our actual sample data, it automatically takes skewing effects into account. Notice that the data from this histogram closely approximate a normal curve and that the values obtained for the mean and SDs are virtually identical to those obtained using the theoretical distribution in . What this tells us is that even though the sample data were indeed somewhat skewed, a -test will still give a legitimate result. Moreover, from this exercise we can see that with a sufficient sample size, the -test is quite robust to some degree of non-normality in the underlying population distributions. Issues related to normality are also discussed further below.

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