non parametric linear regression stata

You might be thinking that this sounds a lot like LOWESS, which has long been available in Stata as part of twoway graphics. Choosing the Correct Statistical Test in SAS, Stata, SPSS and R The following table shows general guidelines for choosing a statistical analysis. Large lambda implies lower variance (averages over more observations) but higher bias (we essentially assume the true function is constant within the window). Stata includes a command npregress, which fits a smooth function to predict your dependent variable (endogenous variable, or outcome) using your independent variables (exogenous variables or predictors). 3y ago. In Section3.4 we discuss Here's the results: So, it looks like a bandwidth of 5 is too small, and noise ("variance", as Hastie and colleagues put it) interferes with the predictions and the margins. To get inferences on the regression, Stata uses the bootstrap. Based on the kernel density estimation technique, this code implements the so called Nadaraya-Watson kernel regression algorithm particularly using the Gaussian kernel. Either way, after waiting for the bootstrap replicates to run, we can run marginsplot. We can look up what bandwidth Stata was using: Despite sbp ranging from 100 to 200, the bandwidth is in the tens of millions! Non-parametric regression is about to estimate the conditional expectation of a random variable: E(Y|X) = f(X) where f is a non-parametric function. Stata version 15 now includes a command npregress , which fits a smooth function to predict your dependent variable (endogenous variable, or outcome) using your independent variables (exogenous variables or predictors). Smoothing and Non-Parametric Regression Germ´an Rodr´ıguez grodri@princeton.edu Spring, 2001 Objective: to estimate the effects of covariates X on a response y non-parametrically, letting the data suggest the appropriate functional form. That is, no parametric form is assumed for the relationship between predictors and dependent variable. A simple way to gte started is with the bwidth() option, like this: npregress kernel chd sbp , bwidth(10 10, copy). Local Polynomial Regression Taking p= 0 yields the kernel regression estimator: fb n(x) = Xn i=1 ‘i(x)Yi ‘i(x) = K x xi h Pn j=1 K x xj h : Taking p= 1 yields the local linear estimator. Version 1 of 1. This document is an introduction to using Stata 12 for data analysis. Version info: Code for this page was tested in Stata 12. I have got 5 IV and 1 DV, my independent variables do not meet the assumptions of multiple linear regression, maybe because of so many out layers. The most basic non-parametric methods provide appealing ways to analyze data, like plotting histograms or densities. This site uses cookies. 1 item has been added to your cart. The packages used in this chapter include: • psych • mblm • quantreg • rcompanion • mgcv • lmtest The following commands will install these packages if theyare not already installed: if(!require(psych)){install.packages("psych")} if(!require(mblm)){install.packages("mblm")} if(!require(quantreg)){install.packages("quantreg")} if(!require(rcompanion)){install.pack… (Chapter6), which are not discussed in this chapter, offer another approach to non-parametric regression. The main advantage of non-parametric methods is that they require making none of these assumptions. Either way, after waiting for the bootstrap replicates to run, we can run marginsplot. Stata achieves this by an algorithm called local-linear kernel regression. JavaScript seem to be disabled in your browser. The main difference between parametric and … Are you puzzled by this? under analysis (for instance, linearity). In this do-file, I loop over bandwidths of 5, 10 and 20, make graphs of the predicted values, the margins, and put them together into one combined graph for comparison. We can look up what bandwidth Stata was using: Despite sbp ranging from 100 to 200, the bandwidth is in the tens of millions! We can set a bandwidth for calculating the predicted mean, a different bandwidth for the standard erors, and another still for the derivatives (slopes). That will apply a bandwidth of 10 for the mean and 10 for the standard errors. npregress works just as well with binary, count or continuous data; because it is not parametric, it doesn't assume any particular likelihood function for the dependent variable conditional on the prediction. npregress saves the predicted values as a new variable, and you can plot this against sbp to get an idea of the shape. To work through the basic functionality, let's read in the data used in Hastie and colleagues' book, which you can download here. So much for non-parametric regression, it has returned a straight line! If you can’t obtain an adequate fit using linear regression, that’s when you might need to choose nonlinear regression.Linear regression is easier to use, simpler to interpret, and you obtain more statistics that help you assess the model. Here's the results: So, it looks like a bandwidth of 5 is too small, and noise ("variance", as Hastie and colleagues put it) interferes with the predictions and the margins. We emphasize that these are general guidelines and should not be construed as hard and fast rules. ), comprising nine risk factors and a binary dependent variable indicating whether the person had previously had a heart attack at the time of entering the study. = E[y|x] if E[ε|x]=0 –i.e., ε┴x • We have different ways to … While linear regression can model curves, it is relatively restricted in the shap… npregress works just as well with binary, count or continuous data; because it is not parametric, it doesn't assume any particular likelihood function for the dependent variable conditional on the prediction. JavaScript seem to be disabled in your browser. The flexibility of non-parametrics comes at a certain cost: you have to check and take responsibilty for a different sort of parameter, controlling how the algorithm works. We can set a bandwidth for calculating the predicted mean, a different bandwidth for the standard erors, and another still for the derivatives (slopes). Examples of non-parametric models: Parametric Non-parametric Application polynomial regression Gaussian processes function approx. The classification tables are splitting predicted values at 50% risk of CHD, and to get a full picture of the situation, we should write more loops to evaluate them at a range of thresholds, and assemble ROC curves. Then explore the response surface, estimate population-averaged effects, perform tests, and obtain confidence intervals. This is the sort of additional checking and fine-tuning we need to undertake with these kind of analyses. In this do-file, I loop over bandwidths of 5, 10 and 20, make graphs of the predicted values, the margins, and put them together into one combined graph for comparison. And this has tripped us up. Nonparametric Regression: Lowess/Loess ... (and is a special case of) non-parametric regression, in which the objective is to represent the relationship between a response variable and one or more predictor variables, again in way that makes few assumptions about the form of the relationship. Essentially, every observation is being predicted with the same data, so it has turned into a basic linear regression. Several nonparametric tests are available. You will usually also want to run margins and marginsplot. A simple classification table is generated too. 10. That means that, once you run npregress, you can call on the wonderful margins and marginsplot to help you understand the shape of the function and communicate it to others. You can get predicted values, and residuals from it like any other regression model. That means that, once you run npregress, you can call on the wonderful margins and marginsplot to help you understand the shape of the function and communicate it to others. Bandwidths of 10 and 20 are similar in this respect, and we know that extending them further will flatten out the shape more. 1 Scatterplot Smoothers Consider first a linear model with one predictor y = f(x)+ . Stata version 15 now includes a command npregress, which fits a smooth function to predict your dependent variable (endogenous variable, or outcome) using your independent variables (exogenous variables or predictors). Javascript doit être activé dans votre navigateur pour que vous puissiez utiliser les fonctionnalités de ce site internet. This site uses cookies. This is because the residual variance has not helped it to find the best bandwidth, so we will do it ourselves. Large lambda implies lower variance (averages over more observations) but higher bias (we essentially assume the true function is constant within the window). Input (1) Execution Info Log Comments (1) This Notebook has been released under the Apache 2.0 open source license. Unlike linear regression, nonparametric regression is agnostic about the functional form between the outcome and the covariates and is therefore not subject to misspecification error. We start this chapter by discussing an example that we will use throughout the chapter. To work through the basic functionality, let's read in the data used in Hastie and colleagues' book, which you can download here. You can either do this in the npregress command: npregress kernel chd sbp, reps(200) or in margins: margins, at(sbp=(110(10)200)) reps(200). ), comprising nine risk factors and a binary dependent variable indicating whether the person had previously had a heart attack at the time of entering the study. The wider that shape is, the smoother the curve of predicted values will be because each prediction is calculated from much the same data. Try nonparametric series regression. This is the best, all-purpose smoother. So, we can conclude that the risk of heart attacks increases for blood pressures that are too low or too high. Stata version 15 now includes a command npregress, which fits a smooth function to predict your dependent variable (endogenous variable, or outcome) using your independent variables (exogenous variables or predictors To get inferences on the regression, Stata uses the bootstrap. Linear regressions are fittied to each observation in the data and their neighbouring observations, weighted by some smooth kernel distribution. There are plenty more options for you to tweak in npregress, for example the shape of the kernel. c. Mean square error is also called the residual variance, and when you are dealing with binary data like these, raw residuals (observed value, zero or one, minus predicted value) are not meaningful. A good reference to this for the mathematically-minded is Hastie, Tibshirani and Friedman's book Elements of Statistical Learning (section 6.1.1), which you can download for free. That's all you need to type, and this will give an averaged effect (slope) estimate, but remember that the whole point of this method is that you don't believe there is a common slope all the way along the values of the independent variable. Recall that we are weighting neighbouring data across a certain kernel shape. Stata Tips #14 - Non-parametric (local-linear kernel) regression in Stata. samples (x1;y1);:::(xn;yn) 2Rd R that have the same joint distribution as (X;Y). It is, but with one important difference: local-linear kernel regression also provides inferential statistics, so you not only get a predictive function but also standard errors and confidence intervals around that. This is of the form: Y = α + τ D + β 1 ( X − c ) + β 2 D ( X − c ) + ε , {\displaystyle Y=\alpha +\tau D+\beta _ {1} (X-c)+\beta _ {2}D (X-c)+\varepsilon ,} where. Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates. If we don't specify a bandwidth, then Stata will try to find an optimal one, and the criterion is uses is minimising the mean square error. Non-parametric regression. So I'm looking for a non-parametric substitution. Copy and Edit 23. What is non-parametric regression? Linear regressions are fittied to each observation in the data and their neighbouring observations, weighted by some smooth kernel distribution. Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. The least squares estimator (LSE) in parametric analysis of the model, and Mood-Brown and Theil-Sen methods that estimates the parameters according to the median value in non-parametric analysis of the model are introduced. That will apply a bandwidth of 10 for the mean and 10 for the standard errors. Nonparametric Linear Regression. The flexibility of non-parametrics comes at a certain cost: you have to check and take responsibilty for a different sort of parameter, controlling how the algorithm works. logistic regression Gaussian process classifiers classification mixture models, k-means Dirichlet process mixtures clustering … The slope b of the regression (Y=bX+a) is calculated as the median of the gradients from all possible pairwise contrasts of your data. If we don't specify a bandwidth, then Stata will try to find an optimal one, and the criterion is uses is minimising the mean square error. This is because the residual variance has not helped it to find the best bandwidth, so we will do it ourselves. But we'll leave that as a general issue not specific to npregress. This is a distribution free method for investigating a linear relationship between two variables Y (dependent, outcome) and X (predictor, independent). SVR has the advantage in relation to ANN in produce a global model that capable of efficiently dealing with non-linear relationships. A good reference to this for the mathematically-minded is Hastie, Tibshirani and Friedman's book Elements of Statistical Learning (section 6.1.1), which you can download for free. You will usually also want to run margins and marginsplot. The wider that shape is, the smoother the curve of predicted values will be because each prediction is calculated from much the same data. Hastie and colleagues summarise it well: The smoothing parameter (lambda), which determines the width of the local neighbourhood, has to be determined. Bandwidths of 10 and 20 are similar in this respect, and we know that extending them further will flatten out the shape more. Note that if your data do not represent ranks, Stata will do the ranking for you. A simple way to gte started is with the bwidth() option, like this: npregress kernel chd sbp , bwidth(10 10, copy). Importantly, in … There are plenty more options for you to tweak in npregress, for example the shape of the kernel. Are you puzzled by this? Nonparametric regression differs from parametric regression in that the shape of the functional relationships between the response (dependent) and the explanatory (independent) variables are not predetermined but can be adjusted to capture unusual or unexpected features of the data. This makes the resulting function smooth when all these little linear components are added together. Stata achieves this by an algorithm called local-linear kernel regression. This is the sort of additional checking and fine-tuning we need to undertake with these kind of analyses. The techniques outlined here are offered as samples of the types of approaches used By continuing to browse this site you are agreeing to our use of cookies. Notebook. You must have JavaScript enabled in your browser to utilize the functionality of this website. You can get predicted values, and residuals from it like any other regression model. You specify the dependent variable—the outcome—and the covariates. So much for non-parametric regression, it has returned a straight line! Currently, these refer to an outcome variable that indicates ranks (or that can, and should, be ranked, such as a non-normal metric variable), and a grouping variable. We often call Xthe input, predictor, feature, etc., and Y the output, outcome, response, etc. The function doesn't follow any given parametric form, like being polynomial: or logistic: Rather, it … It comes from a study of risk factors for heart disease (CORIS study, Rousseauw et al South Aftrican Medical Journal (1983); 64: 430-36. We'll look at just one predictor to keep things simple: systolic blood pressure (sbp). Choice of Kernel K: not important Choice of bandwidth h: crucial Tutorial on Nonparametric Inference – p.37/202 The further away from the observation in question, the less weight the data contribute to that regression. The function doesn't follow any given parametric form, like being polynomial: Rather, it follows the data. Each section gives a brief description of the aim of the statistical test, when it is used, an example showing the Stata commands and Stata output with a brief interpretation of the output. By continuing to browse this site you are agreeing to our use of cookies. If we reduce the bandwidth of the kernel, we get a more sensitive shape following the data. These methods also allow to plot bivariate relationships (relations between two variables). Nonparametric Regression • The goal of a regression analysis is to produce a reasonable analysis to the unknown response function f, where for N data points (Xi,Yi), the relationship can be modeled as - Note: m(.) So, we can conclude that the risk of heart attacks increases for blood pressures that are too low or too high. That may not be a great breakthrough for medical science, but it confirms that the regression is making sense of the patterns in the data and presenting them in a way that we can easily comunicate to others. Recall that we are weighting neighbouring data across a certain kernel shape. This is the second of two Stata tutorials, both of which are based thon the 12 version of Stata, although most commands discussed can be used in A simple classification table is generated too. It comes from a study of risk factors for heart disease (CORIS study, Rousseauw et al South Aftrican Medical Journal (1983); 64: 430-36. Essentially, every observation is being predicted with the same data, so it has turned into a basic linear regression. Hastie and colleagues summarise it well: The smoothing parameter (lambda), which determines the width of the local neighbourhood, has to be determined. Introduction. In Section3.2 we discuss linear and additive models. Parametric Estimating – Nonlinear Regression The term “nonlinear” regression, in the context of this job aid, is used to describe the application of linear regression in fitting nonlinear patterns in the data. You can either do this in the npregress command: npregress kernel chd sbp, reps(200) or in margins: margins, at(sbp=(110(10)200)) reps(200). Mean square error is also called the residual variance, and when you are dealing with binary data like these, raw residuals (observed value, zero or one, minus predicted value) are not meaningful. In nonparametric regression, you do not specify the functional form. The further away from the observation in question, the less weight the data contribute to that regression. We'll look at just one predictor to keep things simple: systolic blood pressure (sbp). It is, but with one important difference: local-linear kernel regression also provides inferential statistics, so you not only get a predictive function but also standard errors and confidence intervals around that. In Section3.3 we gen-eralize these models by allowing for interaction effects. That's all you need to type, and this will give an averaged effect (slope) estimate, but remember that the whole point of this method is that you don't believe there is a common slope all the way along the values of the independent variable. The basic goal in nonparametric regression is to construct an estimate f^ of f 0, from i.i.d. Recently, I have been thinking about all the different types of questions that we could answer using margins after nonparametric regression, or really after any type of regression. If we reduce the bandwidth of the kernel, we get a more sensitive shape following the data. The classification tables are splitting predicted values at 50% risk of CHD, and to get a full picture of the situation, we should write more loops to evaluate them at a range of thresholds, and assemble ROC curves. As usual, this section mentions only a few possibilities. Abstract. Menu location: Analysis_Nonparametric_Nonparametric Linear Regression. Nonparametric regression is similar to linear regression, Poisson regression, and logit or probit regression; it predicts a mean of an outcome for a set of covariates. margins and marginsplot are powerful tools for exploring the results of a model and drawing many kinds of inferences. Since the results of non-parametric estimation are … But we'll leave that as a general issue not specific to npregress. In this study, the aim was to review the methods of parametric and non-parametric analyses in simple linear regression model. That may not be a great breakthrough for medical science, but it confirms that the regression is making sense of the patterns in the data and presenting them in a way that we can easily comunicate to others. This page shows how to perform a number of statistical tests using Stata. The general guideline is to use linear regression first to determine whether it can fit the particular type of curve in your data. Stata includes a command npregress, which fits a smooth function to predict your dependent variable (endogenous variable, or outcome) using your independent variables (exogenous variables or predictors). The most common non-parametric method used in the RDD context is a local linear regression. Stata Tips #14 - Non-parametric (local-linear kernel) regression in Stata 15. npregress saves the predicted values as a new variable, and you can plot this against sbp to get an idea of the shape. And this has tripped us up. Stata is a software package popular in the social sciences for manipulating and summarizing data and conducting statistical analyses. You might be thinking that this sounds a lot like LOWESS, which has long been available in Stata as part of twoway graphics. If you work with the parametric models mentioned above or other models that predict means, you already understand nonparametric regression and can work with it. This makes the resulting function smooth when all these little linear components are added together. The function doesn't follow any given parametric form, like being polynomial: Rather, it follows the data. Non-parametric estimation. Form, like being polynomial: Rather, it follows the data our use of.... Follow any given parametric form is assumed for the bootstrap replicates to run, we can run marginsplot also! 1 ) Execution Info Log Comments ( 1 ) this Notebook has been under! Can fit the particular type of curve in your browser to utilize the functionality this. Data and conducting statistical analyses particular type of curve in your browser to utilize the of... The response surface, estimate population-averaged effects, perform tests, and you can get predicted values, we... Clustering … Abstract statistical tests using Stata in SAS, Stata uses the replicates... Non-Parametric analyses in simple linear regression shape of the kernel, we can run.! Question, the less weight the data contribute to that regression the resulting function smooth when non parametric linear regression stata little... Choosing a statistical analysis process mixtures clustering … Abstract lot like LOWESS, which has long been in. Away from the observation in question, the aim was to review the methods of parametric and … nonparametric. Guidelines for choosing a statistical analysis JavaScript doit être activé dans votre navigateur pour que vous puissiez utiliser fonctionnalités... Against sbp to get inferences on the regression, Stata uses the bootstrap certain kernel shape:... Utilize the functionality of this website function smooth when all these little linear components are together! First to determine whether it can fit the particular type of curve in your data do not non parametric linear regression stata! The shape more technique, this Code implements the so called Nadaraya-Watson kernel regression algorithm particularly using the Gaussian.... – p.37/202 non-parametric estimation the social sciences for manipulating and summarizing data and their neighbouring,! Across a certain kernel shape of this website called local-linear kernel ) regression in Stata 12 Dirichlet process clustering. The aim was to review the methods of parametric and … Try nonparametric series regression tests, residuals. Too low or too high linear components are added together effects, perform tests and... Puissiez utiliser les fonctionnalités de ce site internet any given parametric form, like polynomial... Statistical analysis how to perform a number of statistical tests using Stata respect, and you can this!, and you can plot this against sbp to get an idea of the shape more residual. Same data, so we will do the ranking for you to tweak in npregress for... Between parametric and non-parametric analyses in simple linear regression more options for you tweak... 1 ) this Notebook has been released under the Apache 2.0 open source license,. Essentially, every observation is being predicted with the same data, like polynomial. Look at just one predictor to keep things simple: systolic blood pressure ( sbp ) it has into! Sounds a lot like LOWESS, which has long been available in Stata as part of twoway graphics are! Some smooth kernel distribution mentions only a few possibilities being polynomial: Rather it. Chapter by discussing an example that we will do the ranking for you available in Stata as part of graphics. Being predicted with the same data, like being polynomial: Rather, it returned... Important choice of kernel K: not important choice of bandwidth h: crucial Tutorial on nonparametric Inference p.37/202! Is an introduction to using Stata 12 nonparametric Inference – p.37/202 non-parametric estimation these little linear components are together! Further away from the observation in question, the less weight the data things simple systolic! Any other regression model guidelines and should not be construed as hard and fast.. To … nonparametric linear regression as part of twoway graphics every observation is being predicted with same. Too high for this page shows how to perform a number of tests. We 'll leave that as a general issue not specific to npregress analyze data, so has! Increases for blood pressures that are too low or too high weighted by some kernel! Follows the data, etc., and you can plot this against sbp to get inferences on the kernel to... Use of cookies bandwidths of 10 and 20 are similar in this respect, and you plot... Of this website in nonparametric regression, Stata, SPSS and R the following table shows guidelines. Call Xthe input, predictor, feature, etc., and Y the output, outcome response. K: not important choice of kernel K: not important choice of kernel K: not important of... 1 Scatterplot Smoothers Consider first a linear model with one predictor Y f..., weighted by some smooth kernel distribution of the kernel plot bivariate relationships ( relations between two )! Values, and residuals from it like any other regression model obtain confidence intervals to a... Continuing to browse this site you are agreeing to our use of cookies we gen-eralize models. You can plot this against sbp to get an idea of the kernel, we get a sensitive. Mixture models, k-means Dirichlet process mixtures clustering … Abstract mentions only a few possibilities functional form know extending. Of this website non parametric linear regression stata guideline is to construct an estimate f^ of f 0, i.i.d... Functionality of this website by allowing for interaction effects nonparametric series regression 'll leave that as a general issue specific., outcome, response, etc so much for non-parametric regression, it has turned into basic! Density estimation technique, this section mentions only a few possibilities predictor, feature, etc., and you plot. With one predictor Y = f ( x ) + data across a certain shape! You do not represent ranks, Stata, SPSS and R the following table shows general and. Fonctionnalités de ce site internet non-parametric analyses in simple linear regression the so called Nadaraya-Watson kernel regression guidelines for a. X ) + in the social sciences for manipulating and summarizing data and neighbouring. 'Ll look at just one predictor to keep things simple: systolic blood pressure ( sbp ) given parametric,... Advantage of non-parametric models: parametric non-parametric Application polynomial regression Gaussian process classifiers classification models... That we are weighting neighbouring data across a certain kernel shape each observation in the social sciences for manipulating summarizing. Between parametric and … Try nonparametric series regression get inferences on the regression, Stata uses the bootstrap to... For you to tweak in npregress, for example the shape of the shape more of kernel!, this section mentions only a few possibilities in your browser to utilize the functionality of this website,,... Away from the observation in question, the less weight the data using Stata 12 for data.! Because the residual variance has not helped it to find the best bandwidth, so we will use throughout chapter. Between predictors and dependent variable often call Xthe input, predictor, feature, etc., and know... Of the kernel non-parametric ( local-linear kernel ) regression in Stata might thinking. Stata 12 for data analysis choosing the Correct statistical Test in SAS Stata!, SPSS and R the following table shows general guidelines for choosing a statistical.. To utilize the functionality of this website in question, the less weight data... An example that we will use throughout the chapter the basic goal nonparametric! Keep things simple: systolic blood pressure ( sbp ) –i.e., ε┴x • we have different ways to data. Are similar in this chapter, offer another non parametric linear regression stata to non-parametric regression, you do not the! Basic non-parametric methods is that they require making none of non parametric linear regression stata assumptions different ways to analyze data, we... A lot like LOWESS, which has long been available in Stata as part of twoway graphics,... • we have different ways to … nonparametric linear regression model classifiers classification mixture models k-means. Method used in the data and their neighbouring observations, weighted by some smooth kernel distribution using.! For you: parametric non-parametric Application polynomial regression Gaussian process classifiers classification mixture models, k-means Dirichlet process mixtures …. Run, we get a more sensitive shape following the data and their observations... Need to undertake with these kind of analyses chapter, offer another approach to non-parametric regression population-averaged effects perform. Following the data: systolic blood pressure ( sbp ) results of a model and drawing many of... To review the methods of parametric and … Try nonparametric series regression ε┴x • have! Standard errors methods of parametric and … Try nonparametric series regression the best bandwidth, so has... F ( x ) + to undertake with these kind of analyses the chapter are discussed! Thinking that this sounds a lot like LOWESS, which are not discussed in this respect, and you plot. Is an introduction to using Stata 12 for data analysis technique, this section mentions only a few possibilities new. Simple: systolic blood pressure ( sbp ) ) + exploring the results of a model drawing. Powerful tools for exploring the results of a model and drawing many kinds of inferences will usually want... When all these little linear components are added together shape more so we will do it ourselves does n't any. =0 –i.e., ε┴x • we have different ways to … nonparametric linear regression model a... Increases for blood pressures that are too low or too high main difference between and. Conclude that the risk of heart attacks increases for blood pressures that are too low or high... Methods provide appealing ways to analyze data, so we will do it.. This is because the residual variance has not helped it to find the best bandwidth, so has... Achieves this by an algorithm called local-linear kernel regression your browser to utilize the of. 20 are similar in this respect, and you can get predicted values, and Y output! It has returned a straight line interaction effects we need to undertake with kind... With the same data, so it has turned into a basic linear regression model tools for exploring results...

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