Thanks to the weights, values at low concentrations have a higher influence on where the regression line will be drawn than high concentrations.
Otherwise the high variance at high concentrations could sort of pull the regression line to one or the other direction depending on what kind of a scatter the samples happen to show.
This would deteriorate the bias estimation. Otherwise ordinary linear regression model and weighted least squares model make practically the same assumptions of the statistical qualities of the data set. Error is assumed to be found only in the vertical direction and this error is assumed to be normally distributed.
The constant SD assumption of ordinary linear regression model made it possible to use correlation as a measure of whether the error related to the results is small enough that we can omit the error related to the comparative method in our calculations.
One way to examine the reliability is to switch the comparison direction i. Image 7 shows why switching the comparison direction can have a significant effect on the bias estimation given by weighted least squares.
The assumed true concentrations of samples may change significantly. On the right, S3 in interpreted to have larger concentration than S4. These kind of effects may cause bias estimation to be affected by which method is used as comparative method. In this example, the upper end of the regression line is clearly placed differently into the graphs.
Similar effects are also possible when using ordinary linear regression, though the correlation requirement keeps it rather small. This is the main problem with weighted least squares.
Comparison direction affects the interpretation of sample concentrations. Unlike in ordinary linear regression, the assumed sample concentration affects how a sample is weighted in weighted least squares calculations. There is no way to evaluate how much the error related to the comparative method affects the results.
To get a better idea about what this means, take a look at Image 8 and Image 9. In Image 8 we have plotted the same data set to the regression plots. On the left, method A is our candidate method, while on the right, method B is our candidate method. Regression fit is done using weighted least squares to both of these graphs. Comparing the two graphs, we can clearly see that the regression line has been placed differently into the data set.
On the left, the linear fit goes through the two data points of lowest concentrations S1 and S2. On the right, these dots are clearly below the regression line. A little bit up the concentrations, sample S5 seems to touch the regression line on the left.
When we continue up the concentrations to the next data point that seems to touch the regression line, we find sample S16 that is below the regression line in both graphs. On the right it is assumed that the random error related to method B is also negative in that data point.
As we are calculating bias between methods A and B, these two interpretations are clearly in conflict with each other. Image 9 shows us the regression plots of this same data set with method A as the candidate method. In the middle we see the linear regression fit made by weighted Deming model.
The regression line created by Passing-Bablok on the right is quite concordant with the results of the Deming model, but the confidence interval is significantly wider. So basically weighted least squares is never a reasonable choice unless considering the comparative measurement procedure as a reference happens to suit your purposes.
Deming models take a slightly more complicated approach to linear regression. They make a more realistic assumption that both measurement procedures contain error, making them applicable for data sets with less correlation. Otherwise the reasoning behind these models is quite similar as the reasoning behind ordinary linear regression and weighted least squares.
Both methods are now assumed to contain error, but we are still effectively calculating mean values to estimate bias. To handle situations where one of the measurement procedures gives more accurate results than the other, both Deming models use an estimate of the ratio of measurement procedure imprecisions to give more weight to the more reliable method.
For similar measurement procedures, this ratio is often estimated as 1. Otherwise the assumptions are similar as for ordinary least squares and weighted least squares. Deming models also assume symmetrical distribution. Random errors that cause variance on candidate and comparative measurement procedures are assumed to be independent and normally distributed with zero averages. Deming regression models are very sensitive to outliers. If you use either one of the Deming models, you will need to make sure that all real outliers are removed from the analysis.
In Validation Manager this is rather easy, as on the difference plot Validation Manager shows you which results are potential outliers. After considering whether a highlighted data point really is an outlier you can remove it from the analysis with one click of a button. Clinical data often contains results that are aberrant. The distribution may not be symmetrical. Instead it is mixed, e. In average bias estimation, we had to use median instead of mean for skewed data sets.
Similarly in linear regression, we need to use Passing-Bablok model. Passing-Bablok regression model does not make any assumptions about the distribution of the data points samples nor errors.
It basically calculates the slope by taking a median of all possible slopes of the lines connecting data point pairs. Correspondingly, the intercept is calculated by taking the median of possible intercepts. As a result, there are approximately as many data points above the linear fit as there are below it. Passing-Bablok regression also has the benefit of not being sensitive to outliers.
Image 11 shows one example data set that is too difficult for the other regression models to interpret. Mere visual examination of the regression lines created with weighted least squares upper left corner and weighted Deming model lower left corner raises doubt that these regression lines do not describe the behavior of the data set very well.
Passing-Bablok upper right corner is the model to use in this case, though it is advisable to measure more samples to get more confidence on the linearity of the data and to reach a narrower confidence interval. Study now. See Answer. Best Answer. Study guides. Algebra 20 cards. A polynomial of degree zero is a constant term. The grouping method of factoring can still be used when only some of the terms share a common factor A True B False.
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What does divergent mean? What does mundane mean? What does average joe mean? Here, we see that the standardized residual for a given data point depends not only on the ordinary residual, but also the size of the mean square error MSE and the leverage h ii.
The column labeled " FITS1 " contains the predicted responses, the column labeled " RESI1 " contains the ordinary residuals, the column labeled " HI1 " contains the leverages h ii , and the column labeled " SRES1 " contains the standardized residuals. The value of MSE is 0. Therefore, the first standardized residual The good thing about standardized residuals is that they quantify how large the residuals are in standard deviation units, and therefore can be easily used to identify outliers:.
Using a cutoff of 2 may be a little conservative, but perhaps it is better to be safe than sorry. The key here is not to take the cutoffs of either 2 or 3 too literally.
Instead, treat them simply as red warning flags to investigate the data points further. Let's take another look at the following data set influence2. In our previous look at this data set, we considered the red data point an outlier, because it does not follow the general trend of the rest of the data.
Let's see what the standardized residual of the red data point suggests:. Indeed, its standardized residual 3. We sure spend an awful lot of time worrying about outliers.
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