By using programs for structural equation modeling (SEM) such as lavaan or Mplus (that entail regression models as special cases of SEM), you can place formal constraints on regression parameters and test these constraints by looking at tests of model fit (e.g., a chi-square test). For example, you could restrict (fix) regression intercepts to zero to see if this constraint fits your data. Using SEM programs, you could do this efficiently by using multi-sample (multigroup) analysis in which you can look at all data sets in a single model/run. In multigroup analysis, you can also test equality constraints across groups (e.g., setting regression intercepts and/or slopes equal across groups).

"Are there tests I can use to determine objectively that these two data sets are in error?"

Such a test would require you to compare your data against an objective and fixed criteria. For instance, if I have a record of body weights (in kg, say) and there is a negative value, then I know, objectively, that this value is in error, because zero is a fixed objective lower bound for possible values of "body weight".

If you don't have such a criterion, all you can do is to set up a hypothesis within a statistical model (e.g. you model the intercept as a random variable with a, say, normal distribution; your hypothesis is that the expected value of this distribution is zero), then you can evaluate how (un-)expected your observations are under this model. But this does not tell you anything about whether or not the data are in error. It may just be more plausible that the data is in error when it is very unexpected under a reasonable model.

Can you maybe describe what the theoretical relationship is ? Is it just about the intercept being zero ?

Sal Mangiafico Jochen Wilhelm Christian Geiser

Thank you for taking the time to provide answers to my question. Following Professor Mangiafico’s suggestion, attached are two data plots, one for wheat and one for plastic pellets. If the data matches the theoretical expression, the data should fit the linear expression y = mx + 1. The attachment presents the two data sets whose intercepts are furthest from one. Forcing the trend line through the one for the wheat is visually unacceptable, while the forcing for the plastic pellets appears acceptable visually with only a limited loss of accuracy. On average the mean intercept value for the other 10 data points is 0.993 ± 0.027. The slope varies for all the data sets and is a property of the material being tested: this is in keeping with the theory supporting the expression used.

My view is that the spread in intercept values for the 10 data sets arises from acceptable experimental errors. This quality of agreement verifies that the data are explained by the theoretical expression. This might also be the case for the plastic pellets, but for the wheat something else is awry. As the data are closely grouped around the regression line, it is likely that there is a common cause, possibly a mismeasurement of one of the properties of the wheat.

Currently, I am prepared to accept the 11 data points for further analysis to develop a common expression and reject the data for wheat. This decision is subjective, and I am looking for something more objective, hence the question.

By adjusting one of the wheat properties, I can force the regression line through one. What are your views on doing this? I could either include the wheat in the further analysis or evaluate the 11 data sets and show that the adjusted data for wheat matches the expression developed. Gasps of disapproval?

If I understand correctly, you want to test whether your intercept of the simple linear regression is statistically significantly different from 1.0. You can do this by fitting a constrained regression model where the intercept is not estimated as a free parameter but instead a priori fixed to 1.0. Then, you get a chi-square test of model fit with 1 degree of freedom. If the test is significant, your null hypothesis/assumption that the intercept is equal to 1.0 in the population is rejected.

This is easily done in programs for structural equation modeling (SEM) such as lavaan (free R package) or Mplus (for a simple model like this, even the free demo versions of Mplus or other SEM programs could be used.)

One simple approach is to fit the data to a linear regression, without forcing the intercept to be 1, as you have done.

You have the estimate for the intercept and the standard error for the intercept.

From there, you can 1) Calculate a t-test for the intercept compared to a value of 1. Or 2) Calculate a 95% confidence interval for the intercept, and see if this interval includes the value 1.

This should be straightforward in most software packages.

Beyond that, I'm not sure about your idea if it makes sense to adjust the measured properties of wheat to adjust the trend line to go through an intercept of 1. It may or not make sense for what you want to do. But also think about what it means if you measured this property incorrectly for wheat, does this suggest that you may have measured it incorrectly for the other materials ?

Sal Mangiafico

Thank you for your reply. The 95% CL test eliminates both data sets shown in the attachment, which does not seem reasonable to me. I appreciate your note of caution on adjusting measured parameters.

Gentlemen, Regrettably, both hypothesis test suggestions (chi and t) result in the elimination of more data sets than seems reasonable, at least when viewed through my subjective eye. If I remove the high intercept value of 1.0787, the mean intercept value of the remaining 11 data sets is 1.001 with an SD of 0.0245; consequently, 1.0787 lies more than 3 SDs (1.0745) from the mean. This point falls outside the 99.7% CL and can be considered an outlier and eliminated. Is this approach valid?

I accept that my knowledge of statistics is sufficient for me to be dangerous.

Your data fit pretty tightly around the regression line, so the confidence intervals for the parameters are also pretty tight.

I estimated the model for plastic pellets just from the plot you provided, with the 95% confidence intervals for the parameters. The R code is below. (You could just run this at https://rdrr.io/snippets/ ). The results are pretty close to yours. And, yes, the 95% confidence interval for the intercept does not include 1. Even if you expand this to a 99% confidence interval, the intercept does not include 1.

It's possible to come up with an ad-hoc criterion to determine which models you consider to have an intercept close enough to 1. I'm not sure it makes sense to use the sd across all the intercepts though. You might just formalize your intuitions, identifying those models where the fitted intercept is, say, > 1.05 or < 0.95 .

I think you have to consider what it means for your models if the intercept is not 1. Are the intercepts systematically higher than 1 ? If so, what does this imply ? (I don't know, because I don't understand what theoretical model you are using, or why it's important that the intercept is 1. [You probably don't need to try to explain this; I probably won't really understand.] ).

_____________________

RESULTS FOR APPROXIMATE DATA FOR PLASTIC PELLETS

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.044e+00 1.384e-02 75.44

Sal Mangiafico

WOW, I appreciate the contribution and the time taken to prepare your analysis; above and beyond. With your suggestions and those of Christian Geiser, I feel more confident in removing the high-intercept data set from the secondary analysis. I will refrain from adjusting any measurement parameters to improve the fit as I could be adjusting the wrong parameter, and once adjusted the data set is on a different basis than those not adjusted.

The model is a dimensionless equation of motion describing the performance of a vertical pneumatic conveyor. The value of 1 represents the terminal velocity, the gravitational retarding force, and the slope represents the energy losses arising from particle wall contacts. The greater the error in the intercept value, the greater the error in the slope.

Of the 11 data sets, 6 intercept values are greater than one and five less than one. I estimated the errors associated with each process parameter (air flow rate, solids feed rate, pressure drop, etc.) and can attribute the majority of the variation in intercept values to process measurement errors. The remaining error, I assume is from the errors in measuring the particle properties, particularly the particle terminal velocity. The wheat was subject to degradation during testing and was replaced several times. The particle properties were measured only for the first batch of material. With hindsight, each batch should have been sampled. If the high intercept value arises from a higher-than-determined terminal velocity, additional testing would likely have identified it. The deadly sin of laziness strikes again.

Warm regards, John Wheeldon

prof Salvatore is always a helpful person and I hope all the time keeping well.

Thanks, Abdulaziz Abaid .

Sal Mangiafico

Professor Mangiafico, I note in your answer that you use the 95% CL as the criteria for identifying the outlier intercept values, while others might have used the 99% limits. Are there guidelines governing the selection of which criterion to use?

John M Wheeldon , I don't think there are any real guidelines for this. It's analogous to, in a hypothesis tests, using a criterion of p ≤ 0.05 or p ≤ 0.01. In your case a 99% CL will "include" a wider range of values.

Please prof Salvatore , I have 2 questions , I appreciated a lot if I have answer to my questions?

1- I collected data by 6-likert scale questionnaire of total 48 independent variables to find the manager effectiveness perception as perceived by lecturers in HE. Unfortunately, when I collected the data I have not consider to take any measurement the dependent variables (effectiveness perception). How can I extract the dependent variables from these IDVs ( is it possible to take some items from the 48 already collected and assigned as DV, i.e like competence, performance, empowerment) or take an average of each item and assigned it as DV?

2- In statistical analysis to find correlation and regression between variables, Which approach should I follow (frequentist or Bayesian) ? Is there any distinction between them?

I appreciate your help and thanks a lot

Abdulaziz Abaid , it's probably best to pose this as a new question, with some details about your data and objectives.