Regression - Science topic

Yes and no, k-1 = 3 dummies, should be structured something like this (attached). I've never used minitab, in R it would simply be

model <- lm(Y ~ B + C + D + E + D1 + D2 + D3, data = data)

summary(model)

Well, constrained optimization can do job. That is, the usual linear regression which minimizes the sum of squares (sum of squared errors as an objective function) can be extended with one (or with many constraints) on the variables, i.e., on the slope and intercept. See, e.g.

If you have the counts for both the positive and negative results, logistic regression would be the better approach. The model is slightly complicated because you have multiple centers (locations) within each treatment group, and two time periods.

You could probably do an initial, more simple, analysis by pooling the centers within each treatment, and just looking at the proportions. Maybe just plot the proportions with confidence intervals about the proportions. A plot like this may tell the whole story.

Eigenvectors in Python might seem incorrect due to several potential issues, such as numerical precision errors, incorrect input matrices, or misunderstanding the output format. Eigenvalue problems can be sensitive to the scaling or conditioning of the matrix, and small computational errors may arise in floating-point arithmetic, leading to seemingly incorrect results. It's also important to check if the eigenvectors are normalized or if they are presented as column vectors or row vectors, as the orientation might differ based on the solver or the specific Python package (like NumPy or SciPy) used. Additionally, eigenvectors corresponding to distinct eigenvalues are not unique and can differ by a scalar multiple, which might also lead to apparent discrepancies.

Nuri M. Abduali I'd argue MANOVA doesn't test "... for differences in multiple dependent variables" or at least not in the sense that people think or that is generally useful.

MANOVA creates a weighted average of the various DVs and uses this as its DV. This weighting is opportunistic and atheoretical - its based on the optimal combination in the data (the weighted linear function of DVs that maximizes the variance). So if you repeated the study with new data for 4 DVS the second analysis using the same 4 DVs would have a different linear combination. This creates interpretability and other issues. Most researchers end up falling back on univariate analyses (incorrectly think the MANOVA has added adequate Type I error protection).

See https://psychologicalstatistics.blogspot.com/2021/08/i-will-not-ever-never-run-manova.html

IMO, this is a prime example of what Ronán Michael Conroy recently described (in another thread) as a JUNK QUESTION. What do you hope to learn that you cannot learn from a simple search using Google, Duck-Duck-Go, or some other search engine? (I find it hard to imagine that you have access to ResearchGate, but not to any search engines.)

Shaban Juma Ally, that is why one should use weighted least squares (WLS) regression. When the coefficient of heteroscedasticity is zero - which should not happen - then WLS regression becomes OLS regression.

Metin Dogan Combining Kernel-based Regularized Least Squares (KRLS) with Local Polynomial Regression (LPR) can indeed be an interesting approach, but it largely depends on your research objectives and the type of robustness or validation you are seeking.

KRLS is a non-parametric regression method that excels in capturing complex, non-linear relationships in your data. On the other hand, LPR, especially with tools like lpoly or its advanced version localp, is often used for visualizing trends in the data or performing exploratory smoothing to understand local patterns.

If your goal is to use LPR as a complementary tool for robustness checks or graphing after KRLS, it can be valid, provided you are clear about the specific purpose:

Ultimately, while combining these methods is not inherently invalid, the validity will depend on your methodological rigor and how you communicate the intent behind using both.

thank you Sal Mangiafico

Correlation and regressions can produce "negative results"...

What is your problem?

Kindly mention more details about the paper.

I have the book Statistical methods in medical research by Armitage and Berry

This sounds correct to me.

Oh that's definitely a typo, should be for instance b = 0.167 (or 1.67? Doesn't say standardized or not so impossible to tell)

Yes, the signs of the direct and indirect effects do matter when calculating the total effect in mediation analysis. Here's how the signs influence the total effect:

Breakdown of Effects in Mediation:

How Signs Matter:

Example:

Interpretation:

One way to assess the accuracy of the PLS method is by calculating the figures of merit (RMSEC, RMSEP, only is not enough), using REP or others according to the attached articles. Another very effective way is to compare the predicted data with the values ​​found in a reference analytical technique for your method (i.e. HPLC)

A similar problem of mine got solved by removing all the optional correlation curves in structural model.

There may be a substantial problem (all B’s are approximately zero), the correlation matrix may be inconsistent, or there are too many predictors.

As R² is zero or positive by definition, and Adj.R² is

1-(1-R²)*(N_Cases-1)/(N_Cases-N_Predictors-1), Adj.R² may become negative with large N_Predictors. For instance, if R²=10% and N=100, Adj.R² is 1% with 9 predictors, but -0,1% with 10 predictors. If this is the case, remove irrelevant and/or insignificant predictors.

Inconsistent matrices may be the result of pairwise deletion, when the correlations are not calculated on exactly the same cases. In that case use listwise deletion and/or remove predictors with many missings.

Finally, if R² was near zero to begin with, and so all B’s were near zero as well, you (and especially your client) has to accept that ‘nothing’ was found. But remember the old saying ‘Absence of proof is not proof of absence’. Null-results may be either because there are really none in the population, but also because the research had simply not enough power to detect those results, for instance because of too few cases (note that the ‘alpha’-error is something else than the ‘beta’-error).

Thanks for your help.

It is recommended that a multi-level model be used to account for the hierarchical structure, even in cases where the intraclass correlation coefficient (ICC) is low.

Variable Selection:

LASSO: This method is useful for selecting predictors, but it is essential to ensure that the multi-level structure is considered.

Stepwise: This method is less ideal for multi-level data, but it can be used with caution.

Significance: It is important to evaluate the significance of predictors in the overall model, but it is crucial to avoid overfitting.

Focus on modeling the hierarchical structure, careful selection, and validation to ensure accuracy.

In the context of testing for structural breaks in time series data, the typical approach involves regressing the dependent variable (e.g., DJIA) on one or more independent variables (e.g., S&P 500) and examining for changes in the relationship over time.

To identify structural breaks, one may utilize the Chow Test in instances where a specific breakpoint is suspected, such as in the case of a known event. Alternatively, the Bai-Perron Test may be employed to identify multiple breakpoints without the necessity of prior knowledge regarding their potential occurrence. These tests assist in determining whether and when a significant change in the relationship between variables has occurred over the period from 2003 to 2024.

You can code and arrange the data into various themes. That way, the size of the data is significantly reduced and is more meaningful as well.

It would be worthwhile to start, as Marius Ole Johansen suggests, by becoming curious about the structure of the DV. Do the categories in the variable represent a smaller number of underlying dimensions? Do they form a scale or several scales? You can work on this from the bottom up, using exploratory statistical methods, but also from the top down by using what you know of the situation to identify likely barrier "clusters" that might be treated together.

But I'd explore the DV before you start modelling it, for sure.

The coded equations enable the comparison between the factors in terms of directions and magnitudes of effects by simply comparing the coefficients. on the other hand, the equations in actual terms are not helpful in that comparison while they can be used for the prediction of the responses. So, you can use either the coded or actual equations for predicting a response but you have to substitute for the factors and other regression terms in coded values for the coded equations or in actual values in actual values equations.

Can you show your models and explain the variables? I find it strange to have a R^2 of .99 (at least in social sciences, psychology and biology...)

Why would you want to predict predictors also from your dependent variables? I cannot find a good reason why this would make sense, but again, this may depend on the research field.

How do you came up with the interaction? Dont you have any theory behind the association of the variables?

As you can see, I am quite confused, it would be very helpful to get more information.

A high or low adjusted R square is not necessarily good or bad. Generally you can get a low adjusted R square in cross-sectional data and high adjusted R square in time series analysis. However, in time series analysis, higher adjusted R square with low or insignificant t values indicate multicollinearity problem in regression analysis

, when posting AI-generated content, please label it as such. You might also care to explain how that particular content is meant to be helpful in answering the question that was posted. TY.

Simindokht Kalani, the two main commands for estimating OLS via SPSS are REGRESSION and UNIANOVA (or GLM). The REGRESSION command is older, and does not support factor variables (i.e., categorical variables). However, indicator variables are special, as they have only 2 categories. So you can include them when using REGRESSION. For convenience, though, it is wise to use 0/1 coding for them.

The UNIANOVA command, on the other hand, does support factor variables. In the GUI, categorical variables are called "fixed factors" IIRC, and quantitative variables are "covariates". In the command syntax, factor variables follow the key word BY and covariates follow the key word WITH.

So, to answer your question, yes, you can easily include indicator variables with either REGRESSION or UNIANOVA in SPSS.

No, since your interval includes 0.0 you can't say with a 95% certainty that the regression coefficient is different from 0

Well, the results can be said to be normal in view of the fact that they are similar at one point in time but distinct in the other circumstances.

Specifically, correlation matrix is a diagnostic test or analysis that shows the degree of relationships between the independent variables and the outcome variable(s) both individually and cumulatively. On the other spectrum, the multiple linear regression model is the estimation test or technique of data analysis that examine the impact or effect of the explanatory variables on the outcome variable of the study. The key similarity here is that, they both examine the relationships between the explanatory variables and the explained variable(s). Their major differences is that, while correlation matrix coefficients are used to determine the presence or absence of high correlations among variables, the coefficients in the Multiple Linear Regression (MLR)model are used to ascertain the explanatory power of the independent parameters on the outcome variable of the study.

It should be noted therefore, that a correlation matrix coefficient can serve as a signal for knowing the likely significance level of the coefficients in the MLR model. However, variations normally exist between the two especially regarding their significance level (P-values) and their corresponding signs which could be negative or positive. Therefore, the differences in their correlation coefficients sign (Whether + or -) sign is a normal scenario in a panel data analysis. However, such variations must be backed by strong justifications from both practical, theoretical, and methodological perspectives.

I hope this could be of great help in your data analysis.

To demonstrate temporal effects in regression results:

1. Use time dummy variables to capture seasonality or periodic trends.

2. Include a time trend variable to account for gradual changes over time.

3. Test for interactions between time variables and other factors.

4. Plot the dependent variable over time or residuals against time to visualize trends.

5. Consider lagged effects or autocorrelation adjustments if applicable.

A very high R-squared in panel data regression indicates that the independent variables explain a large portion of the variation in the dependent variable, which is generally good. However, it could also suggest overfitting or issues with data quality or model complexity. Further analysis is needed to ensure the robustness and validity of the results.

Becareful when you are using ratio scale especially for parametric test in any inferetial data analysis. The rule is that the dependent variable must be in ratio scale and must be normally distributed. If the dependent variable is not normally distributed, then adjust the data using either log transformation or other method.

Dear Moi Iannello

really, my personal opininion is that the whole approach does not make any sense at all.

1) The different models test different statistical hypothesis, therefore, even IF all three approaches would show a significant result, this does not mean that the results converge, nor that the general result is robust. You cannot conclude that, its like comparing apples with pears.

2) You state yourself that the DV is ordinal, in that case using an approach for metric variables, even if you account for clustered data, is not optimal and again, tests something different than the Wilcoxon test.

3) Which ordinal approach did you use, logit or probit regression? I never used it with SPSS, therefore, I do not know the options there.

4) Again, you stated yourself, that you did not account for the clustered data with your ordinal approach. How do you think is the result comparable to the other two?

5) You say: "I do know how to do an ordinal regression with clustered data on spss" Did you mean that you dont know how to do it? If yes, why did you do it in the first place, if you already knew that it is the wrong approach? I am really puzzled..... basically, you knew from the beginning that 2 out of 3 analyses are not suitable for your data (one is not for ordinal data and the other does account for clustered data), what do you expect will the results tell you? Garbage in, garbage out.

Without knowing more, I would use a probit regression, to account for the ordinal data structure. The model assumes a normally distributed latent variable (it uses the normal CDF instead of the logistic CDF, but makes the interpretation easier), which seems reasonable, if you already used a OLS regression and thougt it will work. To account for the clustered data (repeated measures), I would use a linear mixed model (aka. multilevel model). I dont know if this is possible in SPSS, but surely in R with the ordinal package (I havent used it so far, but is capable of cumulative linear mixed models and uses the lme4 syntax) or go Bayesian (I would recommend) and use the brms package, which also uses the lme4 syntax.

When providing a rationale for studying the total regression of a tumor, it is crucial to highlight the importance of histology in comprehending the underlying mechanisms and verifying the full regression. Here are few factors you might take into account:

1. Significance of histology: Histology offers intricate insights into the cellular and tissue structure, enabling a thorough evaluation of the tumor and its reaction to treatment. It aids in verifying the lack of remaining tumor cells and evaluating the degree of tumor regression.

2. Treatment response validation: Histological investigation can confirm the clinical response to treatment by verifying the absence of live tumor cells. It is essential to demonstrate the efficacy of the treatment regimen and ensure accurate assessment of treatment outcomes.

3. Mechanistic understanding of regression: Histology can offer valuable insights into the underlying mechanisms that drive tumor regression. This technique enables the evaluation of cellular and molecular alterations occurring in the tumor microenvironment, such as the infiltration of immune cells, the formation of new blood vessels, and programmed cell death. Gaining insight into these pathways can provide valuable guidance for future treatment approaches and contribute to the advancement of individualized medicines.

4. Documentation and reporting: Histological evaluation offers a uniform and unbiased assessment of tumor regression. It enables precise documenting and reporting of therapy response, allowing for comparison with other trials and aiding data analysis for future research.

To summarize, the justification for studying cases where a tumor completely regresses and the necessity of demonstrating histology involves highlighting the significance of histology in verifying the absence of any remaining tumor cells, validating the effectiveness of treatment, comprehending the mechanisms of regression, and guaranteeing precise documentation and reporting of treatment results.

The ARDL (Autoregressive Distributed Lag) model and its related concepts—ARDL Error Correction Model (ECM) and ARDL Long Run Bounds Test—are different approaches used in econometrics to analyze relationships between variables, particularly in the context of time series data. Here’s how they differ:

In summary, ARDL is the general framework for modeling long-run relationships, ARDL ECM incorporates short-term dynamics through an error correction term, and the ARDL Long Run Bounds Test is a diagnostic tool to test for cointegration and determine the appropriate lag structure in the ARDL model. Each serves a distinct purpose in analyzing time series data and understanding the relationships between variables over time.

I assume this question is a modification or elaboration of the question posted here:

If so, please delete the first question.

Second, a time-dependent covariate is an explanatory variable that can change over time, while an outcome variable is a dependent variable. So it is not possible for your outcome variable to be a time-dependent covariate. It would help to clarify things if you posted a small dataset that shows what your data look like. CSV format would be good, as it would be accessible to everyone regardless of what statistical software they use. What software do you use, by the way?

Thank you very much

Hi Klaus, I don't know anything about Amos, but my guess would be that it has to do with what kind of R2 you are looking at.

The classical R2 measures how well your model fits the data it was trained on. This is sometimes called "calibrated" explained variance and, as you say, it can only go up or remain the same by adding additional explanatory variables.

However, this is not true for another type of R2 - the one that measures how well your model generalizes to new data (so called "validated" explained variance). This can be obtained by using the model on a new dataset or by leaving out part of the original dataset (cross-validation) for this purpose. In this case, it is theoretically possible that the R2 decreases by adding a new variable, if the second variable does not bring additional information relevant to explaining the new data (e.g., it might just fit noise in the original dataset rather then some underlying trend).

My guess would be that your software by default runs some kind of cross-validation in your model and therefore R2 you get is in fact the validated explained variance. If the value go down by adding the second variable it might indicate that the new variable does not provide useful information for predicting new data but instead result in overfitting.

You can do it in many ways. PCA is a nice way to gather important parameters. Another way would be to train multiple models with and without specific features and see how that will influence error. Correlations can also help. However, in most cases you need to use your head and see what parameters, why and how are effecting your results. In some cases ANOVA is a nice technique but only if you think and not blindly thrust in results. For example, speed in metres and speed in centimetres are both just speed so using one of them is enough. I know that was stupid example, but it shows the point. Know your data, analyse what impacts results and you will do great. Good luck, hope it will help even a bit.

Somehow it is not clear to my, what you mean with "does not fit"? Could you please provide the output of the whole analysis? I think this would be helpful.

Thank you for the information

You can use econometric methods such as regression analysis, Vector Autoregression (VAR), or Granger causality tests to analyze how macroeconomic variables affect REIT index returns.

Many thanks for your efforts!!! I will try it out as soon as possible and will provide feedback on github!

All the best,

Rainer

import numpy as np

from sklearn.svm import SVC

from sklearn.datasets import make_classification

from sklearn.model_selection import cross_val_score

from pyswarm import pso

# Create a simple dataset

X, y = make_classification(n_samples=100, n_features=5, n_informative=3, n_redundant=2, random_state=42)

# Define the objective function to be minimized by PSO

def svm_pso_loss(params):

C, gamma = params

# Ensure the parameters are positive and within a reasonable range

if C <= 0 or gamma <= 0:

return float('inf') # return infinity if parameters are out of bounds

# Define the SVM classifier

model = SVC(C=C, gamma=gamma)

# Negative mean cross-validated score (we need to minimize it)

neg_accuracy = -cross_val_score(model, X, y, cv=5).mean()

return neg_accuracy

# Set bounds for C and gamma

lb = [0.01, 0.001] # lower bounds of C and gamma

ub = [100, 10] # upper bounds of C and gamma

# Run PSO

best_params, best_score = pso(svm_pso_loss, lb, ub, swarmsize=50, maxiter=100)

print("Best Parameters: C={}, gamma={}".format(best_params[0], best_params[1]))

print("Best Score: {}".format(-best_score))

Hello! I have the same question - did you find out the cause of the problem? I get really weird results when defining my latent variables in the SEM model, but just fine results when I use the aggregated variables.

Thank you Aldo for your suggestion. I can see the general framework.

Cheers!

Yes, you can run a multilevel model without level 2 predictors. This is sometimes referred to as a random coefficient regression analysis. In that analysis, you would simply model potential variability in the level-1 regression intercept and/or slope coefficients across clusters (level-2 units) without explaining that variability by level-2 predictors. This allows you to properly take into account the non-independence that arises from cluster sampling that is shown by your ICC.

Dear Deukaji Gurung

Certainly! Conducting a study on the energy status of construction materials is a fascinating and relevant topic. Here are some reference databases and research articles that you can explore to gather information on energy status, efficiency, and related parameters of construction materials:

When searching these databases and journals, consider using the following keywords and phrases to find relevant articles and research papers:

If age is an ordinal (ordered categorical) variable in your data set, you can simply add that variable as another predictor to your regression model.

Yes of course, adjusting by age will remove the confounding factor of age. Actually, adjusting the model by a confounding factor is one way of two to remove its effect when checking the effect of another variable.

Look at this example:

if the equation of model with only cognitive score as x and the outcome as y is like this:

y = 1 + 5*x1

and equation of a model with only age:

y = 2 + 3*x2

Assuming that both variables have additive effect (which is may not be true)

then the final equation should be something like this:

2*y = 1 + 5*x1 + 2 + 3*x2 = 3 + 5*x1 + 3*x2

and it can be written like this: 2 * y/2 = (3 + 5*x1 + 3*x2)/2

y = 1.5 + 2.5* x1 + 1.5 * x2

And adding more and more variables to the model will definitely affect the coefficients

The other way is by stratification: you can just select homogeneous sub sample and fit a model and so on as I mentioned in the previous comment

There is another way "propensity score matching/ propensity weighted analysis" which is using the same model of the previous regression and adding weights to the patients which can be used in weighted analysis or scoring analysis ( but I don't encourage it in most cases it doesn't work unless you have a large sample )

This in general, but the question remains:

Is there results conclusive? Absolutely not, the only way that can eliminate the confounding factors is controlled randomized trials or a very big sample of the population like big clinical registries or census

In my opinion and from experience most studies group patients like this: ">18", "18-45", "46-65", "> 65"

In your case it deserves a good letter to the editor to criticize and obvious mistake and I would as the authors to send me their data to replicate their results.. there is a very big mistake if they did so.

>> Tooth loss and age are correlated variables and can't be included in the same model as they will produce a collinearity problem, it must do so.<<<

Thank you. You helped a lot.

My problem could be identified by your suggestion since I could find the variable that had the problem.

Do you have any idea about this Error?

ERROR: no solution; program terminated

thanks for your help.

You can use this:

but polynomial regression is very dangerous to use since it doesn't explain anything and it can never be extrapolated.

I have some questions.

1) Was there some treatment (or intervention) between the baseline and followup scores? If so, did all subjects receive it, or only some of them? And if so to that, how were they allocated to intervention vs control?

2) How many colleges are there? If the number is fairly large, it may be preferable to estimate a multilevel model with subjects at level 1 clustered within colleges at level 2.

If the moderation effect is different from zero (i.e., if there is a moderation/interaction effect), the c path would differ for different values of the moderator variable (Z). Consequently, also the total effect would differ for different values of Z.

Well, I don't get a clear picture of the variables you are considering for your study. Since you are considering daily data and you have more than one variable you can either apply non-linear ARDL(s) or multivariate volatility models depending on the objectives of your study. Bear in mind that the pre-tests are highly instrumental to the choice of models.

Best wishes!

Hi Alana Barton It would be good to have more information here to be able to help more, but have you tried a GLM with both years and temperatures included in the model? Perhaps you'd also need to add an interaction effect between temperature and year (as from what you said there's seems to be an interaction). Further explanatory variables could be added to the model if you have measured them.

Ilia, some thoughts on your model. According to your path diagram you have 4 moderator effects. For such a large model, you need a large sample size to detect all moderator effects simultaneously. Do you have a justification for all of these nonlinear relationships?

Some relationships in the path diagram are missing. First, prior knowledge, personal impact, and attitude should be correlated - these are the predictor variables. Second, prior knowledge and personal impact should have direct effects on the dependent variables behavioral intentions and behavior (this is necessary).

As this model is quite complex, I would suggest to start with analyzing the linear model. If this model fits the data well, then I would include the interaction effects one by one. Keep in mind that you need to use a robust estimation method for parameter estimation because of the interaction effects. If these effects exist in the population, then behavioral intentions and behavior should be non-normally distributed.

Kind regards, Karin

There are two different issues here. The first is with regard to step-wise regression, which is a very old-fashioned technique which is no longer widely accepted. Instead, you should indeed use multiple regression.

The other issue is with regard to multicolinearity. Since you predictors will almost certainly be inter-correlated, you will thus have some degree of multicolinearity. But this goes back your wanting to keep the 5 domains separate, since it is their degree of inter-correlation that creates the multicolinearity.

Have you considered using Structural Equation Analysis, or exploratory factor analysis to clarify whether your 5 domains truly are statistically distinct, as opposed to indicators of a single larger domain.

1. participants will have more negative behavioural intentions and attitudes towards children in the chaotic environment than in the calm environment.

--- because there were only two conditions (levels of your factor), you can use a paired t-test (or wilcoxon if nonparametric) to compare the behavioral intentions/attitudes between the calm and chaotic environment where the same participants were subjected to both environments.

2. these differences (highlighted above) will be magnified in participants high in child-oriented perfectionism compared to participants low in child oriented perfectionism.

--- indeed this is a simple linear regression (not multiple one), you can start with creating a new dependent variable (y) as the difference in behavioral intentions/attitudes between the calm and chaotic environment, then you run a regression on the independent variable of a perfectionism score (x).

The interpretations depends on all aspects either positive nor negative.simply advantages and disadvantages .

OLS (Ordinary Least Squares), (2SLS) Two-Stage Least Squares, and (GMM) Generalized Method of Moments, are all statistical methods used in econometrics.

These tools are used in different contexts and under different assumptions, so they do not generally produce similar results.

Yes the formula to obtain a linear T score is quite simple to apply. The question is more about uniform T score or normalized T score in order to address the sweness and kurtosis of different slopes.

The elasticity of y with respect to x is defined as the percentage change in y resulting from a one-percent change in x, holding all else constant. In the context of your regression model, where you have regressed the growth rate of y (which can be thought of as the percentage change in y) against the growth rate of x (the percentage change in x), the estimated coefficient on the growth rate of x is an estimate of this elasticity directly.

Here's why: If you run the following regression:

Δ%y=a+b(Δ%x)+ϵ

where Δ%y is the growth rate of y (dependent variable), Δ%x is the growth rate of x (independent variable), a is the constant term, b is the slope coefficient, and ϵ is the error term, the coefficient b represents the change in Δ%y for a one-unit change in Δ%x. Because Δ%y and Δ%x are already in percentage terms, the coefficient b is the elasticity of y with respect to x.

So, if you have estimated the coefficient b from this regression, you have already estimated the elasticity. There is no need to recover or transform the coefficient further; the estimated coefficient b is the elasticity of y with respect to x.

It's important to note that this interpretation assumes that the relationship between y and x is log-linear, meaning the natural logarithm of y is a linear function of the natural logarithm of x, and the model is correctly specified without omitted variable bias or other issues that could affect the estimator's consistency.

When using efficiency scores as a dependent variable in subsequent regression analysis, researchers often encounter the issue of these scores being bounded between 0 and 1, which violates the assumption of unboundedness in standard linear regression models. To address this issue, fractional regression models, such as the fractional logistic regression, are employed as they are designed specifically for dependent variables that are proportions or percentages confined to the (0,1) interval.

Fractional logistic regression, based on the quasi-likelihood estimation, can be used to model relationships where the dependent variable is a fraction or proportion, which is exactly the nature of technical efficiency scores resulting from DEA. Therefore, it is suitable to apply fractional logistic regression in a two-stage DEA analysis where the first stage involves calculating the efficiency scores, and the second stage seeks to regress these scores on other explanatory variables to investigate what might influence the efficiency of the DMUs.

This two-stage approach, where the DEA is used first to compute efficiency scores and then fractional logistic regression is used in the second stage, helps to avoid the potential biases and inconsistencies that might arise if standard linear regression techniques were used with bounded dependent variables. It is an appropriate statistical technique for dealing with the special characteristics of efficiency scores and can provide more reliable insights into the factors influencing DMU efficiency.

Dear D. S. Chauhan

Here are some freely available books and references on the same topic

1. https://books.google.com/books?id=ZhrIIJat6fgC&pg=PA83&dq=free+book+%27%27MULTIVARIATE+DATA+ANALYSIS:+Using+SPSS+and+AMOS%27%27&hl=ar&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiutLa97tyDAxVf3wIHHSefAWcQ6AF6BAgNEAI#v=onepage&q&f=false

2. https://books.google.com/books?id=dmDez8-ygc4C&newbks=1&newbks_redir=0&lpg=PP1&dq=free%20book%20''MULTIVARIATE%20DATA%20ANALYSIS%3A%20Using%20SPSS%20and%20AMOS''&hl=ar&pg=PP1#v=onepage&q&f=false

3.https://books.google.com/books?id=eH6_ty48As8C&printsec=frontcover&dq=free+book+%27%27MULTIVARIATE+DATA+ANALYSIS:+Using+SPSS+and+AMOS%27%27&hl=ar&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiutLa97tyDAxVf3wIHHSefAWcQ6AF6BAgCEAI#v=onepage&q&f=false

Yes, I have mathematical and statistical knowledge for research. I have also some knowledge of SPSS and R programming for data analysis and visualization

Jiayao Ye

You can see the following books for more details

1. Using Multivariate Statistic by Tabachnic

2. Hair - Multivariate Analysis 7e

3. Applied Multivariate Techniques - Subhash Sharma

Dear Saswati Chutia

Yes, it is possible

Statistical tests such as relationship tests, regression, comparison, etc

Randomness of the sample is not required

Each test has separate special conditions

It can be applied to any block of data that meets the conditions regardless of the type of sampling method performed

However, the sampling method affects the generalizability of the results

best wishes

The assimilation phenomena are the influences that a sound makes to another. We have the progressive assimilation when the preceding sound changes the characteristics of the following: [aθ't̪̟eka] the voiceless interdental fricative consonant interdentalized the voiceless dental oclusive consonant. In the other hand, the regressive assimilation takes place when the following sound influences to the preceding: [an̪dalu'θia] the voiced dental occlusive consonant dentalized the alveolar nasal, but here we have also the progressive assimilation because the alveolar nasal made the voiced dental occlusive consonant mantain its occlusion. Finally, the dissimilation is the process by which two sounds variate their characteristics in order to emphasise the difference between them: an example could be the evolution of the medieval Spanish sibilants.