84 Squared

[Eliz Mettert]

Thatsaid, a squared-up swing does not have to be a hard or fast swing. For example, this Adley Rutschman 95.8 MPH exit velocity (video link) single up the middle was almost perfectly squared up (97%), because the combination of a 77.2 MPH curveball and a below-average swing speed of 67 MPH each worked together to limit the highest possible exit velocity available to 98.7 MPH, which he came close to attaining.

The standard ISO 11146 requires intensity distribution in at least 10 observation planes from which is computed the M squared factor. They are usually acquired using a camera moved along the optical axis. Imagine Optic proposes an innovative approach allowing the acquisition of the complete dataset needed in only one shot.

The Application M2 corresponds to an optimized display of laser quality metrics based on WAVEVIEWTM long-proven optical metrology software. It provides an instant visualization of the M factor and of the intensity map at any observation plane. One can access these values in only one shot

Several mounting options are available:

+ Adaptors for the most common mechanical stages

+ Magnetically coupled top and bottom plates, allowing to mount, remove, and replace CAM SQUARED with a high repeatability

There is no generally agreed upon way to compute R-squared for generalized linear models, such as PROC MIXED. A number of methods have been proposed, these all have certain advantages and certain disadvantages. Your favorite search engine will find many discussions about this.

I chose this formula, R-squared = 1 - SSE_Model / SSE_IntOnly. SSE represents the sum of squared residuals from the model and SSE_IntOnly represents the sum of squared residuals from the intercept-only model. I chose this model because I was looking for a simple and less complicated formula to calculate the percent reduction in variance from the null model to the full model. I used the covariance parameter estimates table from proc mixed to calculate the R-squared. _Design.pdf

I am not sure if I have explained this well! I am very new to calculating the R-squared for multilevel models. I am not sure if this approach is the best or if R-squared should even be calculated this way, but it was a simple formula for me.

I also found this formula, R-squared = SSR/CTSS, where the SSR is the reduction sums of squares due to the model over and above the mean and the CTSS is the corrected total sum of squares. I got the same percent reductions using this formula.

While this works, remind yourself over and over that the sums of squares in a mixed model are NOT what is optimized. It is a maximum likelihood method, and only in the fully balanced design with uncorrelated errors would the sums of squares be the same. A good substitute might be to look at the AIC values and determine the amount of information retained from the null model in the fit model. You could even put this on a relative basis. See the Wikipedia article on Akaike Information Criterion _information_criterion , which is a very good summary and points out how to compare models and the caveats involved.

I actually looked at the AIC as well! Maybe I should just focus on the AIC instead of the pseudo-R-squared because as you have stated the sum of squares is not what is being optimized in mixed models.

The Kramer paper looks quite good, and I can see some utility in the MLE based pseudo-R2. However, you would have to be sure to change to an ML method from the standard REML methods used in MIXED and GLIMMIX, and that leads to biased estimates (as a simple example, compare the biased estimate of the variance (denominator=n) to the unbiased estimate (denominator = n-1), the proof that the biased estimate is the ML estimate is a pretty standard math stats course proof). I think we are still looking for an appropriate approach to goodness of fit for REML mixed models.

Is there a way to quickly create the sum of revenue squared as a calculated measure? Measuring confidence on calculated measures, so getting access to the data to preform this function in Excel - is proving difficult.

Linear regression calculates an equation that minimizes the distance between the fitted line and all of the data points. Technically, ordinary least squares (OLS) regression minimizes the sum of the squared residuals.

Before you look at the statistical measures for goodness-of-fit, you should check the residual plots. Residual plots can reveal unwanted residual patterns that indicate biased results more effectively than numbers. When your residual plots pass muster, you can trust your numerical results and check the goodness-of-fit statistics.

R-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression.

The regression model on the left accounts for 38.0% of the variance while the one on the right accounts for 87.4%. The more variance that is accounted for by the regression model the closer the data points will fall to the fitted regression line. Theoretically, if a model could explain 100% of the variance, the fitted values would always equal the observed values and, therefore, all the data points would fall on the fitted regression line.

In some fields, it is entirely expected that your R-squared values will be low. For example, any field that attempts to predict human behavior, such as psychology, typically has R-squared values lower than 50%. Humans are simply harder to predict than, say, physical processes.

Furthermore, if your R-squared value is low but you have statistically significant predictors, you can still draw important conclusions about how changes in the predictor values are associated with changes in the response value. Regardless of the R-squared, the significant coefficients still represent the mean change in the response for one unit of change in the predictor while holding other predictors in the model constant. Obviously, this type of information can be extremely valuable.

No! A high R-squared does not necessarily indicate that the model has a good fit. That might be a surprise, but look at the fitted line plot and residual plot below. The fitted line plot displays the relationship between semiconductor electron mobility and the natural log of the density for real experimental data.

The fitted line plot shows that these data follow a nice tight function and the R-squared is 98.5%, which sounds great. However, look closer to see how the regression line systematically over and under-predicts the data (bias) at different points along the curve. You can also see patterns in the Residuals versus Fits plot, rather than the randomness that you want to see. This indicates a bad fit, and serves as a reminder as to why you should always check the residual plots.

This example comes from my post about choosing between linear and nonlinear regression. In this case, the answer is to use nonlinear regression because linear models are unable to fit the specific curve that these data follow.

However, similar biases can occur when your linear model is missing important predictors, polynomial terms, and interaction terms. Statisticians call this specification bias, and it is caused by an underspecified model. For this type of bias, you can fix the residuals by adding the proper terms to the model.

While R-squared provides an estimate of the strength of the relationship between your model and the response variable, it does not provide a formal hypothesis test for this relationship. The F-test of overall significance determines whether this relationship is statistically significant.

On Thursday, October 15, 2015, a disbelieving student posted on Reddit: My stats professor just went on a rant about how R-squared values are essentially useless, is there any truth to this? It attracted a fair amount of attention, at least compared to other posts about statistics on Reddit.

It turns out the student's stats professor was Cosma Shalizi of Carnegie Mellon University. Shalizi provides free and open access to his class lecture materials, so we can see what exactly he was "ranting" about. It all begins in Section 3.2 of his Lecture 10 notes.

In case you forgot or didn't know, R-squared is a statistic that often accompanies regression output. It ranges in value from 0 to 1 and is usually interpreted as summarizing the percent of variation in the response that the regression model explains. So an R-squared of 0.65 might mean that the model explains about 65% of the variation in our dependent variable. Given this logic, we prefer our regression models to have a high R-squared. Shalizi, however, disputes this logic with convincing arguments.

One way to express R-squared is as the sum of squared fitted-value deviations divided by the sum of squared original-value deviations: $$R^2 = \frac\sum (\haty - \bar\haty)^2\sum (y - \bary)^2 $$ We can calculate it directly using our model object like so:

1. R-squared does not measure goodness of fit. It can be arbitrarily low when the model is completely correct. By making \(\sigma^2\) large, we drive R-squared towards 0, even when every assumption of the simple linear regression model is correct in every particular.

What is \(\sigma^2\)? When we perform linear regression, we assume our model almost predicts our dependent variable. The difference between "almost" and "exact" is assumed to be a draw from a normal distribution with mean 0 and some variance we call \(\sigma^2\).

Shalizi's statement is easy enough to demonstrate. The way we do it here is to create a function that (1) generates data meeting the assumptions of simple linear regression (independent observations, normally distributed errors with constant variance), (2) fits a simple linear model to the data, and (3) reports the R-squared. Notice the only parameter for sake of simplicity is sig (sigma). We then "apply" this function to a series of increasing \(\sigma\) values and plot the results.

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