Original link:tecdat.cn/?p=6335
On 18 September 2014, residents of Scotland will vote in a referendum on whether to become independent from the United Kingdom. Since a series of polls were conducted last week with slightly different results, I decided to do a simple meta-analysis. Based on available evidence, findings, and estimates of current status.
data
Including the last six votes from September 14, the website gives data from whatscotlandThinks.org. As shown below.
| Polls open | End of the poll | The number of voters | agree |
|---|---|---|---|
| September 9, 2014 | November 9, 2014 | 1205 | 571 |
| September 9, 2014 | November 9, 2014 | 820 | 400 |
| September 9, 2014 | November 9, 2014 | 992 | 475 |
| October 9, 2014 | December 9, 2014 | 844 | 389 |
| October 9, 2014 | December 9, 2014 | 642 | 345 |
| October 9, 2014 | December 9, 2014 | 943 | 466 |
Analysis of the
Analyze with R. This function combines the number of respondents to give an overall estimate of the average percentage of voters who agree. Two analyses were performed, one called a fixed effect analysis and the second a random effect analysis.
The results of
The forest map below shows the results of the analysis. The overall estimated percentage of votes (under the random-effects model) was 48.72%, with a 95% confidence interval of 46.82% to 50.62%. So, based on these six polls, the estimate is that the yes vote is less than 50%, while the confidence interval shows that the data is consistent with the “true” vote, which is higher than 50%.
The 95% confidence intervals from different studies overlap with each base sample point, illustrating that with a (relatively) small number of points given in each study, small differences in results may be due purely to sampling error.
The simple analysis presented here has many flaws.
A meta-analysis is like a random sample. In fact, poll samples are constructed using more sophisticated survey design methods and probably shouldn’t be analyzed as I did (as if they were simple random samples).
R output
For those interested, the R output of the analysis shown below gives the proportion for each poll, the 95% CI for each poll, the weight assigned to each poll (in the fixed and random effects analysis), and an estimate of I ^ 2 (the proportion of variation attributable to true heterogeneity).
Proportion 95%-CI %W(fixed) %W(random) The Times and The Sun 0.4739 [0.4453; 0.5025] 22.12 19.16 The Guardian 0.4878 [0.4531; 0.5226] The Observer 0.4788 [0.4473; 0.5104] 18.23 17.57 Better Together 0.4609 [0.4269; 0.4952] The Sunday Telegraph 0.5374 [0.4979; 0.5765] 11.75 13.94 The Sunday Times 0.4942 [0.4618; 0.5266] 17.36 17.16 Number of studies combined: K =6 PROPORTION 95%-CI Z p.Value Fixed effect Model 0.4859 [0.4726; 0.4991] NA -- Random effects model 0.4872 [0.4682; 0.5062] NA -- Quantifying heterogeneity: Tau ^2 = 0.0045; H = 1.42 [1; 2.25]; I^2 = 50.3% [0%; 80.3%] Test of heterogeneity: Q D.F. P. value 10.07 5 0.0734 Details on analytical method - Inverse variance method - DerSimonian-Laird estimator for tau^2 - Logit transformation - Clopper-Pearson confidence interval for individual studiesCopy the code
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