Odit molestiae mollitia laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio voluptates consectetur nulla eveniet iure vitae quibusdam? Excepturi aliquam in iure, repellat, fugiat illum voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos a dignissimos. Close Save changes. Sample variance generally gives an unbiased estimate of the true population variance, but that does not mean it provides a reliable estimate of population variance.
Here, I show that sample variance itself has high variance at low sample sizes. I run through a variety of empirical simulations that vary population size and population variance to see what general patterns emerge.
Our goal is to figure out how reliable smaller samples are with respect to estimates of variance. Each row number will correspond to its sample size. This process is repeated reps times for each sample size. Reps will span the columns.
The red circles represent the pooled mean differences for the same samples using our estimation formulas we connected the corresponding symbols for clarity. As seen from the Figure 2 , the estimates of the mean were fairly accurate and useful. However, in some situations, using these estimates might still be better than the alternative — excluding the trials which reported the wrong summary data median instead of mean. Using our estimation method, we can see the effect of such trials on pooled summary measures.
In the next section we will illustrate our method in an actual systematic review. The results were expressed as the mean increase in hemoglobin in Epo arm compared with the control. However, a number of the papers reported median increase instead of mean increase and standard deviation. Due to lack of available methods to use median values, the authors of this important review, decided not to use these papers in their meta-analysis. Recently, the Cochrane review was published attempting to provide more updated analysis of the effects of Epo in anemia related to malignancy [ 5 ].
The Cochrane reviewers did meta-analyze data to calculate an average weighted mean increase in hemoglobin as the result of Epo treatment. However, the Cochrane investigators could not include the totality of evidence in relation to this outcome since a number of the trials reported data as medians instead of means. Therefore, published meta-analyses related to the effect of Epo in anemia due to malignancy suffer from the phenomena akin to the outcome reporting bias [ 6 ] simply due to fact that methods are not yet developed to allow researches to use data medians.
Here we illustrate that it is actually possible to use medians and pool, and improve inclusiveness of meta-analyses. Their results show that on average Epo increases hemoglobin by 2. However, the Cochrane investigators could not pool data from other available studies in the literature with similar eligibility. However, they did not report the data for the standard deviation of these means.
Since the size of each arm is 15 patients, our formula 16 provides the best estimate of the standard deviation using the median and the range. We used Figure 1 on page in Welch at al. Thatcher et al do report in their paper ranges of hemoglobin for patients treated by Epo and control.
This trial was a three-arm study, in which two doses of Epo were compared against the control. For the purpose of this analysis, we separated the data from each of the Epo arms and compared them against one half of the control group just like the rest of the studies in the Cochrane review.
When we incorporated these results into the Cochrane meta-analysis, we found that the effect of Epo on mean increase in hemoglobin significantly changed: the pooled estimate decreased from an average of 2. An example: Meta Analysis with all eligible trials included. Using our estimation formulas, we were able to include two other studies eligible for this meta-analysis [9, 10]. The pooled estimate decreased to 1.
Our estimates come with some uncertainty. The summary pooled estimate now ranged from the low of 1. This example outlines how our method can be potentially useful for meta-analysts. It is important to realize that this example is provided only to illustrate our method. Nevertheless, we believe that this example is a good illustration of the potential of our method. While it is common practice that the investigators simply pool what is available to them it is actually not known how often studies are excluded because of reporting a different summary statistic.
In future we will attempt to systematically address this issue and evaluate, for example, how often the Cochrane reviews did not pool data from the available median values when they pooled data on continuous outcomes. We hope that availability of our methods to the wider meta-analytic audience may further improve the inclusiveness of all relevant studies for the Cochrane and other meta-analyses.
Using simulation methods we were able to determine that formula 5 is a best estimator for the mean when dealing with a small sample size. As soon as sample size exceeds 25, the median itself is the best estimator. Using simulations, we determined that for very small samples up to 15 the best estimator for the variance is the formula In summary, the best estimators for the mean and the standard deviation for different sample sizes are given in Table 3. Google Scholar.
Petiti DB: Meta-analysis, decision analysis and cost-effectiveness analysis. Methods for quantitative synthesis in medicine. J Clin Oncol. Cochrane Database Syst Rev. Int J Clin Oncol. Cancer J Sci Am. S: Epoetin alpha prevents anaemia and reduces transfusion requirements in patients undergoing primarily platinum-based chemotherapy for small cell lung cancer.
Br J Cancer. Suppose our sample mean of 3, grams came from a random sample of babies. Means from samples this large did not vary much.
We marked this sample result in a histogram for samples of size From advanced probability theory, we have a probability model for the sampling distribution of sample means. The model reinforces what we have already observed about the center and gives more precise information about the relationship between sample size and spread.
0コメント