☰ MedicalStat

Weighted Estimate


Section 1

For input, that require logarithmic transformation (RR, OR, HR, IRR, ... etc.)

Explanations & examples:
Here it's possible to calculate the weighted estimate (weighted average) between two or more estimates that require log transformation, in other words estimates such as odds ratios (OR), risk ratios (RR), incidence rate ratios (IRR) ... etc., and also to compare two estimates with each other to see if they could be equal. If the estimates are not already log transformed before they are entered into the table, then choose the input option "No" under "Are the input values already log transformed". If the inputs are already log transformed, choose "Yes". Non-log transformed values must be entered as the estimates and their 95 % confidence intervals. Log transformed values must be entered as the log of the values and the standard error of the log transformed values.

Two of the estimates entered can be compared to see if they are equal by calculating their ratio and the difference of their log transformation. If the number 1 is included in the 95 % confidence interval of the ratio, then the null hypothesis H0 can not be rejected, namely that the ratio is 1, and there is no significant difference between the two estimates. If 1 is included in the 95 % confidence interval of the ratio then 0 is included in the 95 % confidence interval of the difference between the log transformed estimates. If, however, 1 is not in the 95 % confidence interval of the ratio, then H0 is rejected and the estimates are significantly different from each other on a five percent significance level.

It is only recommended to use and interpret the weighted average of the estimates entered when they are not significantly different from each other. If they are not significantly different, then the weighted estimate will be the common value of all the estimates combined into one. In the calculation of the weighted estimate the estimates involved will contribute with their "weights" meaning that "heavier" estimates (with more persons in the study) will contribute more, so that the weighted estimate will be closer to the "heavier" of the involved estimates. For more info and the formulas used in the calculations, please see the page medical statistics formulas.

Example:

In a study(a) the OR value 1.29 [1.05 : 1.57] was determined in a logistic regression of the connection between poor eyesight and experiencing a fall among elderly. In another, similar study, the OR value was determined to be 1.61 [1.32 : 1.96]. We want to compare the two OR values to see if they can be assumed equal and, if so, calculate the weighted estimate between them. First of all, we see that neither of the two OR values is included in the 95 % confidence interval of the other one. If this had been the case, we could already then establish, that the OR values were not significantly different. And if there had been no overlap at all between the confidence intervals, this would have established that the OR values were significantly different. In this case, however, with some overlap between the intervals, it's impossible to conclude anything from the intervals alone.

Two OR values being entered for comparison and calculation of their weighted estimate

The comparison and calculation of the weighted estimate between two OR values

Entering the OR values and their confidence intervals into the table, we get a ratio between them of 0.8012 with a 95 % confidence interval of [0.6044 : 1.0623]. Since 1 is included in the 95 % CI of the ratio, it cannot be rejected that the ratio could be 1, and the OR values are therefore not significantly different from each other on a five percent significance level. We could also have done the test instead and looked at the confidence interval. The z-value in the test is 1.5401 and the corresponding p-value 0.1235 which is more than 0.05, so we don't reject the null hypothesis. Since the OR values are not significantly different from each other, we can combine them into one by calculating their weighted estimate, which is 1.4439 with 95 % CI [1.2541 : 1.6626]. The weighted estimate, in this case, is the effect that visual impairment has on the odds of falling, having adjusted for the different studies. The interpretation of the weighted OR of 1.4439 is that if you take two elderly people, who are equal regarding everything else (i.e. also from the same study), where one has visual impairment and the other hasn't, then the visually impaired will have 1.4436 times higher odds of experiencing a fall than the non-impaired. The weighted estimate is significant since 1 is not included in its 95 % confidence interval. So visual impairment does have a significant effect on the odds of falling, having adjusted for different studies.

(a) Yip J et al.: Visual acuity, self-reported vision and falls in the EPIC-Norfolk Eye study. British Journal of Opthamology 2014;98:377-382
Are the input values already log transformed?
Yes No

No. of strata:

Stratum Description Estimate 95 % Lower Bound 95 % Upper Bound SE(ln(estimate))       W       W × Ln(estimate) Relative weight
1
2
Sum

Results

Compare two strata:
Log Transformed Not Log Transformed
Compare Strata Ln(estimate) SE(Ln(estimate)) % Confidence Interval
Estimate % Confidence Interval
A:  
B:  
Estimate SE(Ln(estimate)) % Confidence Interval
Null Hypothesis (H0) Z Value P value
Difference \( \ln(E_A) - \ln(E_B) \) Difference
Ratio \( \frac{E_A}{E_B} \) Ratio
Weighted Estimate of all Strata:
WE SE(Ln(WE)) % Confidence Interval
Null Hypothesis WE Z Value P Value
Ln transformed

Ln(WE)
Not Ln transformed

WE

Decimals:

Section 2

For input, that DON'T require log transformation (Means, MD, RD, ... etc.)

Explanations & examples:
Use this section to calculate the weighted estimate (weighted average) of estimates that do NOT require log transformation in the process, for ex. mean values, mean differences etc. As in the section above it is only advisory to use and interpret the weighted estimate if the estimates involved are not significantly different from each other. The weighted estimate will then be the common estimate of all the groups, combined into one. The weighted estimate will lean more towards the "heavier" of the estimates involved (the estimates from the groups with more people in), since they will "weigh" more in the calculations.

For more info about the weighted estimate and the calculations behind, please see the page medical statistics formulas.

Enter the inputs into the table according to what info you have; if you know the estimates and their standard error, choose the option "standard error". If you have the estimates and their 95 % confidence intervals, choose the option "95 % CI".

A comparison between two estimates can be made by looking at the difference between the estimates (one minus the other) and the 95 % confidence interval of the difference. If the number 0 is included in the 95 % confidence interval of the difference, then it cannot be rejected that the estimates are equal on a five percent significance level. In this case the p-value in the test of equality will be above 0.05. Otherwise the p-value in the z-test will be below 0.05 and the null hypothesis is rejected. The null hypothesis H0 says that the two estimates are equal (there is no significant difference between them). Alternative to looking at the 95 % confidence interval of the difference, the z-test can also be performed directly to get the exact values of z and p and then determine whether the p-value is above or below 0.05. The default null hypothesis H0 is "H0: difference = 0" (i.e. H0:there is no difference between the estimates). But an alternative null hypothesis can also be tested, for ex. "H0: difference = 2.5".

Example:

58 male and 43 female students had their PEFR lung capacity measured. The mean of the males was 568.2 (l/min) and the mean of the females was 474.1 (l/min) with standard errors 8.284 and 7.482 respectively. We want to compare the two means and calculate their weighted estimate if there is no significant difference between them.

Two mean values being entered into the app for comparison and test for equality

Two OR values being entered for comparison and calculation of their weighted estimate

First of all it is noticed, that the two 95 % confidence intervals of the mean values do not overlap each other. If there's no overlap between the intervals we can already now reject the null hypothesis H0, that states that the two mean values are the same (there's no significant difference between them). So it isn't necessary to do a comparison by calculating their difference and its 95 % confidence interval to check for equality. If you wish to do so anyway, the difference of the means is 94.1 with 95 % CI [72.2212 : 115.9788]. As can be seen, 0 is not in the confidence interval of the mean, confirming that the means cannot be the same (the difference can not be assumed 0). So the p-value in the z-test will be below 0.05. If desired, the z-test can be performed with the z-value and corresponding p-value being z = 8.4299 and p = 0.0000. Since H0 is rejected in this case, the means can not be assumed equal and it is therefore not recommended to combine the two means into one by calculating and interpreting their weighted estimate.
What is known?
Standard Error 95 % CI

No. of strata:

Stratum Description Estimate SE 95 % CI       W       W × estimate Relative weight
1
2
Sum

Results

Compare two strata:
Compare Strata Estimate SE % Confidence Interval
Null Hypothesis Difference Z Value P Value
Diff
Difference:
Weighted Estimate of all Strata:
WE SE % Confidence Interval
Null Hypothesis WE Z Value P Value


WE