Multiple Linear Regression - Science topic

In deciding between simple linear regression (SLR) and multiple linear regression (MLR), these are some of the technicalities or factors you need to consider:

*Simple Linear Regression (SLR):*

1. Single independent variable (predictor)

2. Linear relationship between independent and dependent variables

3. Few data points (small sample size)

4. Simple model interpretation

*Multiple Linear Regression (MLR):*

1. Multiple independent variables (predictors)

2. Complex relationships between independent and dependent variables

3. Larger sample size

4. More accurate predictions (if relevant variables are included)

*Additionally, ask yourself some of these questions:*

1. How many independent variables do you have?

- SLR: 1 variable

- MLR: 2 or more variables

2. What is the complexity of the relationship between variables?

- SLR: Simple, linear relationship

- MLR: Complex, non-linear relationships or interactions between variables

3. How many data points do you have?

- SLR: Small sample size (e.g., <100)

- MLR: Larger sample size (e.g., >100)

4. How accurate do you need the predictions to be?

- SLR: Reasonable accuracy for simple models

- MLR: Higher accuracy for complex models

5. Do you need to control for confounding variables?

- SLR: No

- MLR: Yes

*Also, consider this decision Flowchart:*

1. Do you have only 1 independent variable?

- Yes: SLR

- No: Proceed to 2

2. Do you have a simple, linear relationship?

- Yes: SLR

- No: MLR

3. Do you have a small sample size (<100)?

- Yes: SLR

- No: MLR

4. Do you need high accuracy or control for confounding variables?

- Yes: MLR

- No: SLR

*Further Considerations:*

1. Multicollinearity: If independent variables are highly correlated, MLR may not be suitable.

2. Overfitting: MLR can suffer from overfitting if too many variables are included.

3. Model interpretability: SLR is generally easier to interpret than MLR

If the purpose of your study is exploration, as you say, why do you perform hypothesis tests at all? How did you come to these hypotheses you tested in your exploratory study? Usually, you would take the data to generate (ideas about) hypotheses you may test in an independent set of data (ideally collected more specifically with the aim to test just that hypothesis).

David L Morgan , if I had sufficiently good information, I wouldn't do an experiment. If I need to do an experiment, my knowledge is by definition insufficient. To plan the experiment, I need to make a guess on what I expect from the experiment. Of course will I use the best sources of information available, to make a sensible, educated guess. But I must make a guess. So I say: we have to rely on guesswork (to plan a study), but only rarely on pure guesswork.

If these are the only variables in your equation (i.e., there are no covariates), then it sounds like you could perform a 2 x 3 Analysis of Variance.

Simple linear regression analysis (SLRA) that CBR value decreases with increase in plasticity index and also increases with increase in maximum dry density. From SLRA coefficient of correlation R2 for plasticity index and maximum dry density are 0.72 and 0.78 respectively. The soaked and unsoaked CBR approximate values range between 3-9% and 14-38% respectively, Soaked CBR value of the subgrade is usually preferred for designing flexible pavements as it gives the CBR strength of subgrade soil under worst case scenario of a pavement being submerged under water for a minimum period of 4 days during floods in addition to the compaction process, preparation usually involves soaking each specimen in water for 96 hours before the penetration test. The CBR value obtained at 2.5mm penetration is normally higher than that at 5mm penetration and this is the value used

In my research, there are hypotheses, to test them I used multiple linear regression, and after that I want to write a prediction equation in the dependent variable.

According to this equation: Y=A+B1X1+B2X2+,........

Yes, it's generally okay to use a percentage as a dependent variable in multiple linear regression. However, there are some considerations to keep in mind:

Dear ,

Thank you very much for your explanations!

However, I am working on IBM SPSS.

Do you know how I could do a regression within subject design with clustered standard error on IBM SPSS ?

Since my dependent variables are ordinal I thought about doing an ordinal regression. Analyze > Generalized Linear Model > then I could specify the type of model as ''ordinal logistic''. But does this test take into account within subject design?

I also thought about GEE on spss. But I don't know if I can do it. ?

My study :

112 participants had to face 7 different scenarios ( the factors) and indicate their level of agreement on a scale likert scale from strongly disagree to strongly agree to measure their motivation, satisfaction, collaboration and help. I also checked for age, gender, and professional situation.

Thank you in advance

The only difference of note is that the random effects model could be non-linear (e g in analysing binary or count data). The mixed effects model includes a 'fixed part' - that is means, and a 'random part' (that variances and covariances).

Lot of resources here

and the free online course

Of course, you can use multiple linear regression. The results will show which variables have a stronger effect on your dependent variable after controlling for childhood trauma. Considering mediation analysis and attachment styles research, I can suggest that you should consider whether you measure the two dimensions of the attachment scale of romantic relationships or attachment styles as a classification. The second one is whether your sample is clinical or healthy. In this case, if you had not preregistered your study, you can try to analyze the effect of attachment styles or dimensions on personality via childhood trauma after controlling for family functioning. You can use JASP, adding attachment dimensions as independent variables, personality as the dependent variable, and childhood trauma as the mediator, with family functioning as a background confounder. At first, you can use it without a background confounder, and in the second model, you can use it with a background confounder. But I think if you had used romantic attachment styles, attachment styles could be independent variable, family functioning as the mediator, personality as the dependent variable, and childhood trauma as a moderator

Best regards,

Thanks for the additional info, Svetlana Bondar. Just to remind everyone, earlier, you wrote:

"The residuals of the dependent variable in my case are not normally distributed – a bimodal distribution."

I don't remember ever running into this problem myself, but a bit of digging around suggests that one possible cause of this is failure to include an important dichotomous explanatory variable. Do any possibilities come to mind?

Another suggestion I've seen that might be helpful is to use quantile regression rather than OLS regression, and to perhaps model the 25^th and 75^th percentiles instead of (or in addition to) the median.

Finally, I found this article, which seems to be addressing the same problem. Maybe you'll find some good suggestions in it.

https://www.redalyc.org/pdf/899/89949526005.pdf

HTH.

Here are some specific examples of literature on bootstrapping vectors:

I hope this helps!

The p value for the overall F statistic indicates that the overall multiple correlation R (and R squared) value is not significantly different from zero at the .05 level (p is greater than .05). The reason for the non-significant result may be a small sample (as indicated by your degrees of freedom) and/or the presence of too many predictor variables. This also explains the large discrepancy between R squared and adjusted R squared. You are lacking statistical power to "detect" an effect.

you are welcome

The same, just that there is no "*when all other IVs are held constant*".

It's simply the (expected) change in Y per unit change of X. There are no other variables involved that would need to be "held constant".

Hello Pablo José Arana Barbier, Twenty years ago I did generalised linear modelling in SPSS [sorry can't recall the version of the program for sure]. Possibly version 14. This way you can enter multiple factors, continuous variables into the formulae.

Thank you David!!

Presumably, strong significant relationship means the p-value is well below your chosen alpha level (likely 0.05), whereas weak significant relationship means it is just below alpha. Is that right?

Did your instructor give any guidance (or restrictions) on what the sample size is supposed to be for your model? I ask because of the relationship between n and p-values: The larger n gets, the smaller p becomes (all else being equal).

PS- Daniel Wright, Catherine Strutz could have stated explicitly that this question relates to a course assignment (assuming it does--and I think it does, judging by the wording), but I do not think she was deliberately attempting to conceal that fact. YMMV.

You say "...(for example, "residency duration from 0 to 18 years of age" means how long these 65-year-old individuals lived in rural areas during the ages of 0-18)." Sorry. You are correct that for multiple regression they would all be included. So if it was zero years, you count them in that sample. But there are probably fewer zeroes for the number of years lived by a 65-year-old in the 0-18 category, as I explained above, perhaps providing more useful data for that category. Zeroes may be problematic here. Regression involving counts may be done differently, and not in my expertise, but might be useful here.

By the way, are your 65-year-olds currently residents anywhere, or are you only looking at 65-year-old current rural residents (because that would make a difference)?

The only other comment that occurs to me is that I think you want to be careful not to make too much of your results. You might even want to separate your study into different parts with different models, perhaps.

Multiple linear regression (MLR) is a commonly used statistical technique for modeling the relationship between a dependent variable and several independent variables. However, MLR has certain assumptions that must be met for the results to be valid. When these assumptions are violated, the results of the analysis can be biased or misleading.

Here are some common assumptions of MLR:

If one or more of these assumptions are violated, there are several things you can do:

It is important to note that there is no one-size-fits-all solution for dealing with violated assumptions in MLR. The approach you take will depend on the violation's specific nature and your dataset's characteristics. It is important to carefully examine the results of your analysis and to use good judgment in interpreting them.

Bootstrapping is a resampling technique that can be used to estimate the variability of a statistical estimator, such as the coefficients of a multiple linear regression model. The basic idea of bootstrapping is to repeatedly draw samples from the original dataset with replacement, and then compute the statistic of interest (in this case, the coefficients of the regression model) for each sample. By doing this many times, we can obtain a distribution of the statistic, which can be used to estimate its standard error and construct confidence intervals.

To interpret the results of a bootstrapped multiple linear regression, you should start by examining the point estimates of the coefficients, as well as their standard errors and p-values. These can be used to determine the significance of each predictor variable in the model, and to make inferences about the relationship between the predictors and the outcome variable.

Next, you should examine the distribution of the coefficients obtained from the bootstrap samples. This can be visualized using a histogram or a boxplot. If the distribution is roughly symmetric and centered around the point estimate, this suggests that the estimate is stable and reliable. On the other hand, if the distribution is skewed or has long tails, this may indicate that the estimate is more variable and less trustworthy.

Finally, you can use the bootstrap distribution to construct confidence intervals for the coefficients. These intervals represent the range of values that the coefficient is likely to fall within, based on the data and the statistical model. A 95% confidence interval, for example, would include the true value of the coefficient with 95% probability, assuming that the model assumptions are correct. If the confidence interval does not include zero, this suggests that the corresponding predictor variable is significantly related to the outcome variable

It is not the fact that the data are time series which is important. What is important is the presence of autocorrelation. For example, if your data are about the composition of the soil along a road, for instance, every 10 meters, you will also face autocorrelation despite the fact that the data are not time series. It is known how to handle autocorrelation for regression data but it adds difficulties and it requires typically longer samples.

The normality assumption in linear regression concerns the residuals (error terms), not the independent or dependent variables.

Simple and multiple linear regression are statistical techniques used to model the relationship between a dependent variable (also called the response variable) and one or more independent variables (also called predictors or explanatory variables).

Simple linear regression involves modeling the relationship between the dependent variable and a single independent variable. The goal is to find a linear equation that best describes the relationship between the two variables. This equation can be used to make predictions about the value of the dependent variable for a given value of the independent variable. The equation for simple linear regression is:

Y = a + bX + e

where Y is the dependent variable, X is the independent variable, a is the intercept (the value of Y when X is equal to 0), b is the slope (the change in Y for a one-unit change in X), and e is the error term (the difference between the actual and predicted values of Y). The goal is to estimate the values of a and b that minimize the sum of the squared errors.

Multiple linear regression, on the other hand, involves modeling the relationship between the dependent variable and two or more independent variables. The goal is to find a linear equation that best describes the relationship between the dependent variable and all the independent variables together. The equation for multiple linear regression is:

Y = a + b1X1 + b2X2 + ... + bnXn + e

where Y is the dependent variable, X1, X2, ..., Xn is the independent variables, a is the intercept, b1, b2, ..., bn are the slopes, and e is the error term. The goal is to estimate the values of a, b1, b2, ..., bn that minimize the sum of the squared errors.

In both simple and multiple linear regression, the quality of the fit can be assessed using various measures, such as the R-squared value, which indicates the proportion of variance in the dependent variable that is explained by the independent variable(s), or the F-test, which compares the fit of the regression model to a null model (i.e., a model with no independent variables).

Linear regression can be used for various purposes, such as predicting future values of the dependent variable, identifying the most important predictors of the dependent variable, or testing hypotheses about the relationship between the dependent and independent variables. However, it is important to be aware of the assumptions underlying linear regression, such as the linearity, independence, normality, and homoscedasticity of the errors. Violations of these assumptions can lead to biased estimates and invalid conclusions.

How large is your sample size, and how large is the correlation between the two predictors?

Do not average, but use a multi-level model with company-level random effects. Also, you should examine how time itself influences your outcome directly. In one of the R regression engines (lme4, rstanarm, brms), the model would look like this:

outcome ~ 1 + time + pred_1 + pred_2 + (1 + time + pred_1|Company)

In this model, all companies get their own slope over time. Pred_1 is also estimated on populatio-level and per individual company. Pred_2 is an example of a pure population-level effect (aka fixef effect). That is useful for predictors that do not vary within one company, such as the country of residence.

To estimate multi-level models, you'll have to bring your data into long-format (one outcome measure per row), e.g.:

CompID | Pred_1 | Pred_2 | time | outcome

Hope that helps, if not: https://schmettow.github.io/New_Stats/mlm.html

Roshdi R Khalil Amjed Zraiqat Eman Nouh Eman I. El-Awady

Dear Fiseha, when in presence of a multiple linear regression, the usual way of representing it in a bivariate space is to use as axes :

X axis = expected value of Y from the fitted equation

Y axis = observed value of Y.

This representation will give rise (if the equation has a decent fit) of the cloud of points elongated along the main diagonal of the plot (Y=X) allowing you to estimate by eye ouutliers, corresctly fitted points and so forth...

Here is a playlist of videos of how to conduct Meta-Analysis & Meta-Regression In R: https://www.youtube.com/playlist?list=PLrLWLaG7yx85X7ZjN4ySllDQ-hiB9Q4nm

Dear Lina Naji

You may look at the following article.

Best of luck

It's obviously going to be difficult to try to display all those variables in a single plot.

A) A simple approach is to plot the observed values vs. the predicted values from the model.

B) You could plot the dependent variable vs. one continuous independent variable grouped by the categorical variable. Three plots like this may be informative (e.g. : https://rcompanion.org/rcompanion/images/e_04_04.jpg )

Bivariate screening of candidate predictors for a multivariable regression model is considered a bad practice that tends to produce overfitted models. See Mike Babyak's 2004 article, for example:

And having non-significant regressions in a multivariable model is not problematic either. There is more likely a problem when all variables are statistically significant, in fact--unless n is quite large. See these two sections in the DataMethods.org author checklist, for example (link below):

HTH.

Daniel Wright Unfortunately I can no longer see what you had written as Rgate will not let me see previous replies; so I am remembering/guessing.

By 'corrective' action I mean using a method that does not make the strong assumptions of the standard model. For example if you see heteroscedasticity in a catch-all plot I would model that heterogeneity and see if it makes a difference to the result; I think that is much better than some test for it . As Box (1953) notes of Bartlett's test "To make the preliminary test on variances is rather like putting to sea in a rowing boat to find out whether conditions are sufficiently calm for an ocean liner to leave port!" The GAM approach allows for non-linearity, displays the partial residuals so you can see what is going on (very helpful in the multiple predictor case) and gives you the linear model if there is not strong evidence for a curve.

I think we are all getting more aware of the "Forking paths" in the practice of modelling. I am reminded of the phrase that you would not eat sausages if you saw them being made! The good thing about GAM is the inbuilt cross validation which is aimed at reducing overfitting. I have enjoyed reading Stuart Ritchie's Science Fictions and think the only way is to have a plan, being open about what you have done and use cross validation to guard against implicit p-hacking.

Back to the original question - it looks to be an underlying linear relation to me and I would stick with that.

اذا لم نقوم باستعمال الادوات الاحصائية الدقيقية لهذا المجال بالنسبة للمتغيرات البحث سوف يؤدي الى الى اخطاء احصائية في دقة النتائج ولكن ممكن استخدام في بحثين منفصلين كمتغير تابع @

This could be due to a suppressor effect. A suppressor effect can occur when predictor variables are highly correlated with one another. In that case, one predictor can "suppress" variance in the other predictor and thereby enhance the predictive value of that other predictor. It is even possible that a predictor that is not significantly correlated with the DV turns out to be a "significant" predictor in a multiple regression due to suppression.

If you do a literature search for "suppression" or "suppressor effect" in regression, you should be able to find a number of relevant resources.

Again, there is no way that a dichotomous variable could have a non-linear relationship with another variable. Therefore, no need or possibility to check for non-linearity.

You can see this by plotting the dichotomous variable against the DV in a scatter plot. The dichotomous variable has only two possible values on the x axis. The OLS regression line will go through the group means. You could not fit anything other than a straight line through the two means.

Look at the severity of the violation. It may be just negligible given the large sample size.

Add the function of time as another IV just like you would any other eg an interaction term. David Booth

Thou shalt not dichotomize...

There is really no good reason to dichotomize a continuous moderator as this can introduce all kinds of problems. Simply use the continuous version and plot the interaction as suggested by David Booth.

Julie Schipper -

For heteroscedasticity, even after a transformation - which I would only do if necessary, and not to address heteroscedasticity - there can still be heteroscedasticity. I recall an example among those that Penn State puts on the internet where a transformation had been done for purposes of addressing heteroscedasticity in a real estate problem, and they still had substantial heteroscedasticity remaining. You can have a kind of artificial, I say "nonessential" heteroscedasticity, which may be the result of a problem with your model, but the natural, I say "essential" heteroscedasticity that you should expect can be impaired, again, by a problem with your model. Data quality issues can also be involved, etc. See https://www.researchgate.net/publication/354854317_WHEN_WOULD_HETEROSCEDASTICITY_IN_REGRESSION_OCCUR.

To get an idea of your level of heteroscedasticity, see https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

With enough data, you can measure it well using the following:

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

Defaults are mentioned in one sheer of that Excel file, and it would be reasonable for the coefficient of heteroscedasticity as defined here to be from 0.5 to 1.0. See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as determined by Ken Brewer.

For more information, see the following:

https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and the various updates and references there. 

If you are working with data which have already been transformed, then that changes things in ways perhaps both good and problematic. The problematic can also include problems of interpretation, as Jochen noted. I suggest only using transformations if you absolutely must, even if you are going to "back-transform" as Salvatore mentioned. I think that the simpler you can keep things, the better.

Best wishes - Jim

Ask your analyst to explain it coz nobody else says stuff like this. Good luck on finding a better analyst

Best wishes David Booth

Doing a meaningful analysis is, unfortunately, not that simple.

The problem are possible interactions between variables that may either obscure non-linear relationships with the response variable or wrongly imply a non-linear relationship if the impact of the other variables is not considered at the same time (i.e. when you look at the marginal relationships). There is no easy way out. You need to make some assumptions, and they should be resonable - and justifying this is not done by means of statistics but by expert knowledge of the subject matter. A help - to a certain limited amount - can be residual diagnostics: if the residuals do not show any pattern and seem to be a sample from a normal distribution, then this is a slight hint that the model might be approriate or at least that it does not miss anything really important (like an important interaction or non-linear relationship).

Stepwise variable selection is a very bad approach to build a model. You should definitely avoid this. The selection of variables (and non-linear relationships, and interactions) should be based on theoretical considerations, expert knowledge of the subject matter and the purpose of the model. If you consider a variable important enough to inlcude it in the model, then there should not be a statistical reason (like the lack of a statistically significant effect estimate given your data) to exclude it from the model.

If you have many variables, and you are just looking if they allow you to somehow predict the outcome, modelling using linear models (or alikes) may actually not be the best choice. You may consider regression trees, random forrests or deep learning approaches. It's evident that this needs quite a lot of data, but this is the proce you have to pay if you have no theoretical concept but still want to create/build some tool that uses data to make good predictions.

And, finally, there is no BEST MODEL. There are many different possible models, all with their own pros and cons. Only if you have very, very specific demands what the model should do (and if you are able to formulate this in a mathematical way) you may be able to identify a specific model that is best suited for exactly this purpose. And again, decinding on this best model is not (only) a matter of statistics you calculate for your observed data - it remains mainly a task requiring expert knowledge in the subject matter.

If you are expert in the subject matter, it may be very helpful to cooperate with a local statistician to identify a good solution.

Thank you very much for all your answers! I have unfortunately no choice but to use R... so I am going to go for the lme4 method.

Thanks again to you all!

The multiple regression result shows you that there is partial redundancy between the independent variables (they are correlated and overlap in terms of "explaining" the DV). Openness is no longer a significant predictor once the other predictors are in the model because one or more of them accounts for (partially) the same portion of variance as does openness.

Another issue could be overfitting. With additional correlated predictors, standard errors will become larger, reducing statistical power. Therefore, with more predictors in the model, previously significant predictor variables may become insignificant.

The question is what you want to find out. If your goal is to determine the most relevant predictor variables then the multiple regression model would be preferred because it accounts for the redundancy of the predictors. Depending on your sample size, you may not have enough power though given the number of predictors.

Ho Nguyen Tuong -

Are you sure you just want "yes" or "no" for obesity? How about a BMI value? Also, how much smoking? At any rate, you can compare model results for a given sample using a "graphical residual analysis." (You can research that online.) Also, you probably want to consider a "cross-validation," as you do not want to overfit a model to a particular sample when that may mean your model won't fit so well to other data which you wanted to cover. Trying more than one sample could help.

Best wishes - Jim

I'd normalize the DV (that is, rescale it so that the value will be with 0...1) and use a beta-model or a quasi-binomial (regression) model.

Cross validation algorithm in scikit-learn of Python is working very well in tuning hyper-parameters.

I'm confused about the model you are trying to use... Is there one value for Y and several values for X and for B ? And then you need to determine the best value for Co ? ...

Ellen Kroon, an alternative would be to estimate/use Heteroskedasticity Robust Standard Errors in your significance testing. This can be done easily in Stata or R. See the link below for more: https://www.r-econometrics.com/methods/hcrobusterrors/

Try following reference

Thank you very much and hope you are staying warm, Xingyu Zhou

Hello Ayyu,

Reporting conventions differ by outlet, so you may wish to consider what is usual and customary for your target.

If you're talking about summary statistics for the variables involved (means, SDs, correlations among variables), then yes; one table can suffice.

If you're talking about the outcomes of the two methods, then no, generally not. The exception would be if the table was restricted to: (a) test name; (b) p-value; and (c) indicator of whether result was statistically significant.

For MLR, you'd generally want to present: (a) regression coefficients (both unstandardized and standardized) for each independent variable; (b) test of significance for that variable's coefficient; and in a summary, the overall R-square and test of whether the overall regression was significant.

For t-test, a table really isn't needed, beyond that for summary statistics. In text, something like: "t(42) = 4.22, p < .001" can suffice.

Good luck with your work.

Christer Thrane thank you for your responses. I appreciate your help

Hi Hassan Farooq

you may include all of your control variables - age, gender, work experience and education level - in one block and begin the regression analysis with them. Next, you include the independent variables that are supposed to have an effect on your dependent variable. I would recommend entering them one after another so that you can observe the change each new variable causes to the ones already included and the explained variance in the dependent variable. Once you have entered all variables, you will have controled for the initial four variables, presumably rendering they influence on the dependent variable insignificant.

I do not have and article present, but perhaps someone else can help you out.

Best

Marcel

If R2 for your regression is negative, it means that your regression predicts worse than the simple mean value predictor (e.g., when you simply predict that y = mean(x)).

For "graphical residual analyses," you could compare different models on the same scatterplot for the same sample. (Different models here would have one predictor, or the other, or both, with or without an intercept if that is in question, though subject matter knowledge could tell you if y is zero when the predictor or predictors are all zero.) For "cross-validation," you could try this on different subsamples, but results for other data not collected could still be different.

Yufan Ye -

Have you looked at a "graphical residual analysis?" You can search on that term if you aren't familiar. It will help you study model fit, including heteroscedasticity. Also, a "cross-validation" may help you to avoid overfitting to the sample at hand to the point that you do not predict so well for the rest of that population or subpopulation which you wish to be modeling.

If this model is a good fit, I expect you will likely see heteroscedasticity. See https://www.researchgate.net/publication/354854317_WHEN_WOULD_HETEROSCEDASTICITY_IN_REGRESSION_OCCUR.

Plotting your data may be very helpful, and much more informative and practical than "assumption tests."

Cheers - Jim

The units in which the dependent and independent variables are measured do not have to be the same for multiple linear regression. For example, you could try to predict job success in salary dollars (DV) from IQ scores (IV1), number of years of education (IV2), and gender (IV3). In this example, all variables would have different units of measurement.

What is the size of your data? Is it big data or a small one? MLR is a parametric model which assumes certain criteria. Neural needs big data as it uses multiple and many parameters. That may be one reason for this difference. Pls read more about parametric vs non-parametric models

Dear Angel Tabullo

I would suggest using the pool.r.squared() function from the mice package.

It should give your desired results if you are using the lm modelling function.

Best wishes,

Francesco

There will be problems because you used an auto-selection process.

You can fit a given model to obeserved data, but you cannot use observed data to specify the model that should be fitted to. Model specification should be driven by theory. Surely, observed data may give you ideas about an improved model specification ("exploratory data analysis"), and such an improved model can in future be fitted to new observed data.

Automated model selection is a kind of extreme exploratory analysis, you actually lose completely the possibility to interpret any statistical significance of the data. This is in principle nothing bad. Just don't interpret significance and everything is "fine". However, the automated process is clearly not recommended because it ineviably will just find the best fit to all peculiarities of our observed set of data, disregarding any theory, what is never a very clever thing to do.

Yep, would agree, and recommend to use residualised change scores.

Non-normality will not matter much, outliers will.

See these links:

Hi Bram Ten Barge

In statistics, it's crucial to think about testing the assumptions that go along with the analysis and determining the suitability of the data set for various analyses. Using factor analysis, few factors are extracted from the large number of related variables to a more manageable number, prior to using them in other analysis such as multiple regression or multivariate analysis of variance:

All the fellows have contributed well. Whenever we are to analyze the data, we ask ourselves whether we want to check whether variables are related or one is said to have impact on the other. If the case is of measuring strength of relation then we can use cross-tabulation (nominal or ordinal data), Spearman (ordinal data) and Pearson f(Continuous data).

But if the question is regarding one variable being the reason for the other then we need to use regression.

In regards to experience, there are several ways to look at it. Many believe that an inverse correlation between skill level (which is closely aligned with experience) and injury exists--which, based on my research and many others, favors that point of view. However, on the other hand, since the most experienced athletes will be on the field or will play more than their inexperienced peers, the opportunity for injury exposure would be greater. Lastly, more experienced players may be more prone to attempt riskier maneuvers during play, increasing the potential for injury.

Time series analysis allows you to have various models like auto regressive etc.which would explain the behavior of your data based on which you can project the d.v. into the future.

Hi,

Categorical variables as I.Vs can be converted into dummy variables and studied.

Bruce Weaver, thank you for this clarification. Greatly appreciated.