Glm.se software

Notes The names of the columns in the new data frame should exactly match the names of the columns in the data frame that were used to build the model. Published by Zach. View all posts by Zach.

Next The Difference Between glm and lm in R. Leave a Reply Cancel reply Your email address will not be published. The following are permitted: '"independence"', '"exchangeable"', '"ar1"', '"unstructured"' and '"userdefined"'. Type of standard error to be calculated. Defualt 'san. Other options are 'jack': if approximate jackknife variance estimate should be computed. Clusters with size one must not be represented in zcor.

Halekoh, U. Journal of Statistical Software, 15, 2, ". Liang, K. In addition, the figure indicates the existence of two outliers dots in the boxplot. The histogram in the lower left panel shows that, although the mean amount of money spent on presents is Finally, we will plot the amount of money spend on presents against relationship status by attraction in order to check whether the money spent on presents is affected by an interaction between attraction and relationship status.

The boxplot in the lower right panel confirms the existence of an interaction a non-additive term as men only spend more money on women if the men single and they are interested in the women. If men are not interested in the women, then the relationship has no effect as they spend an equal amount of money on the women regardless of whether they are in a relationship or not. We will now start to implement the regression model. In a first step, we create two saturated models that contain all possible predictors main effects and interactions.

The two models are identical but one is generated with the lm and the other with the glm function as these functions offer different model parameters in their output. After generating the saturated models we can now start with the model fitting. Model fitting refers to a process that aims at find the model that explains a maximum of variance with a minimum of predictors see Field, Miles, and Field , In this section, we will use a step-wise step-down procedure that uses decreases in AIC Akaike Information Criterion as the criterion to minimize the model in a step-wise manner.

This procedure aims at finding the model with the lowest AIC values by evaluating - step-by-step - whether the removal of a predictor term leads to a lower AIC value. We use this method here just so that you know it exists and how to implement it but you should rather avoid using automated model fitting. The reason for avoiding automated model fitting is that the algorithm only checks if the AIC has decreased but not if the model is stable or reliable.

Thus, automated model fitting has the problem that you can never be sure that the way that lead you to the final model is reliable and that all models were indeed stable. Imagine you want to climb down from a roof top and you have a ladder. The problem is that you do not know if and how many steps are broken.

This is similar to using automated model fitting. In other sections, we will explore better methods to fit models manual step-wise step-up and step-down procedures, for example. The AIC is calculated using the equation below. The lower the AIC value, the better the balance between explained variance and the number of predictors. AIC values can and should only be compared for models that are fit on the same data set with the same number of cases LL stands for logged likelihood or LogLikelihood and k represents the number of predictors in the model including the intercept ; the LL represents a measure of how good the model fits the data.

The penalty of the BIC is bigger than the penalty of the AIC and it includes the number of cases in the model LL stands for logged likelihood or LogLikelihood , k represents the number of predictors in the model including the intercept , and N represents the number of cases in the model.

Interactions are evaluated first and only if all insignificant interactions have been removed would the procedure start removing insignificant main effects that are not part of significant interactions. Other model fitting procedures forced entry, step-wise step up, hierarchical are discussed during the implementation of other regression models. We cannot discuss all procedures here as model fitting is rather complex and a discussion of even the most common procedures would to lengthy and time consuming at this point.

It is important to note though that there is not perfect model fitting procedure and automated approaches should be handled with care as they are likely to ignore violations of model parameters that can be detected during manual - but time consuming - model fitting procedures. As a general rule of thumb, it is advisable to fit models as carefully and deliberately as possible. We will now begin to fit the model.

The automated model fitting procedure informs us that removing predictors has not caused a decrease in the AIC.

The saturated model is thus also the final minimal adequate model. We will now inspect the final minimal model and go over the model report. The first element of the report is called Call and it reports the regression formula of the model. Then, the report provides the residual distribution the range, median and quartiles of the residuals which allows drawing inferences about the distribution of differences between observed and expected values.

If the residuals are distributed non-normally, then this is a strong indicator that the model is unstable and unreliable because mathematical assumptions on which the model is based are violated. Next, the model summary reports the most important part: a table with model statistics of the fixed-effects structure of the model. The table contains the estimates coefficients of the predictors , standard errors, t-values, and the p-values which show whether a predictor significantly correlates with the dependent variable that the model investigates.

All main effects status and attraction as well as the interaction between status and attraction is reported as being significantly correlated with the dependent variable money. An interaction occurs if a correlation between the dependent variable and a predictor is affected by another predictor.

The top most term is called intercept and has a value of To exemplify what this means, let us consider what the model would predict that a man would spend on a present if he interested in the woman but he is also in a relationship.

The amount he would spend based on the model would be This means that the intercept represents the predicted value if all predictors take the base or reference level. And since being in relationship but being interested are the case, and because the interaction does not apply, the predicted value in our example is exactly the intercept see below. Now, let us consider what a man would spend if he is in a relationship and he is not attracted to the women. In that case, the model predicts that the man would spend only Below the table of coefficients, the regression summary reports model statistics that provide information about how well the model performs.

The difference between the values and the values in the coefficients table is that the model statistics refer to the model as a whole rather than focusing on individual predictors. The multiple R 2 -value is a measure of how much variance the model explains. A multiple R 2 -value of 0 would inform us that the model does not explain any variance while a value of.

A value of 1 would inform us that the model explains percent of the variance and that the predictions of the model match the observed values perfectly. Multiplying the multiple R 2 -value thus provides the percentage of explained variance. Models that have a multiple R 2 -value equal or higher than. It has been claimed that models should explain a minimum of 5 percent of variance but this is problematic as it is not uncommon for models to have very low explanatory power while still performing significantly and systematically better than chance.

In addition, the total amount of variance is negligible in cases where one is interested in very weak but significant effects. It is much more important for model to perform significantly better than minimal base-line models because if this is not the case, then the model does not have any predictive and therefore no explanatory power. The adjusted R 2 -value considers the amount of explained variance in light of the number of predictors in the model it is thus somewhat similar to the AIC and BIC and informs about how well the model would perform if it were applied to the population that the sample is drawn from.

Ideally, the difference between multiple and adjusted R 2 -value should be very small as this means that the model is not overfitted. If, however, the difference between multiple and adjusted R 2 -value is substantial, then this would strongly suggest that the model is unstable and overfitted to the data while being inadequate for drawing inferences about the population.

Differences between multiple and adjusted R 2 -values indicate that the data contains outliers that cause the distribution of the data on which the model is based to differ from the distributions that the model mathematically requires to provide reliable estimates.

The difference between multiple and adjusted R 2 -value in our model is very small Now, we compare the final minimal adequate model to the base-line model to test whether then final model significantly outperforms the baseline model. The comparison between the two model confirms that the minimal adequate model performs significantly better makes significantly more accurate estimates of the outcome variable compared with the baseline model.

After implementing the multiple regression, we now need to look for outliers and perform the model diagnostics by testing whether removing data points disproportionately decreases model fit. To begin with, we generate diagnostic plots. The plots do not show severe problems such as funnel shaped patterns or drastic deviations from the diagonal line in Normal Q-Q plot have a look at the explanation of what to look for and how to interpret these diagnostic plots in the section on simple linear regression but data points 52, 64, and 83 are repeatedly indicated as potential outliers.

The graphs indicate that data points 52, 64, and 83 may be problematic. We will therefore statistically evaluate whether these data points need to be removed.

In order to find out which data points require removal, we extract the influence measure statistics and add them to out data set. The difference in row in the data set before and after removing data points indicate that two data points which represented outliers have been removed.

NOTE In general, outliers should not simply be removed unless there are good reasons for it this could be that the outliers represent measurement errors.

If a data set contains outliers, one should rather switch to methods that are better at handling outliers, e. One alternative would be to switch to a robust regression see here. However, here we show how to proceed by removing outliers as this is a common, though potententially problematic, method of dealing with outliers.

As we have decided to remove the outliers which means that we are now dealing with a different data set, we need to rerun the regression analysis. As the steps are identical to the regression analysis performed above, the steps will not be described in greater detail.

After rerunning the regression analysis on the updated data set, we again create diagnostic plots in order to check whether there are potentially problematic data points. Although the diagnostic plots indicate that additional points may be problematic, but these data points deviate substantially less from the trend than was the case with the data points that have already been removed. To make sure that retaining the data points that are deemed potentially problematic by the diagnostic plots, is acceptable, we extract diagnostic statistics and add them to the data.

The new diagnostic plots do not indicate outliers that require removal. With respect to such data points the following parameters should be considered:. There should not be any autocorrelation among predictors. This means that independent variables cannot be correlated with itself for instance, because data points come from the same subject. If there is autocorrelation among predictors, then a Repeated Measures Design or a hierarchical mixed-effects model should be implemented instead.

Predictors cannot substantially correlate with each other multicollinearity see the subsection on multi- collinearity in the section of multiple binomial logistic regression for more details about multi- collinearity. Indeed, even VIFs of 2. NOTE However, multi- collinearity is only an issue if one is interested in interpreting regression results! If the interpretation is irrelevant because what is relevant is prediction!

See Gries for a more elaborate explanation. Except for the mean VIF value 2. We will now test whether the sample size is sufficient for our model.

The test statistics ranges between 0 and 1 where lower values are better. If the values approximate 1, then there is serious concern as the model is not reliable given the sample size. In such cases, unfortunately, the best option is to increase the sample size. The function smplesz reports that the sample size is insufficient by 9 data points according to Green As a last step, we summarize the results of the regression analysis.

As we have seen above, and is also shown in the table below, the correct R 2 values are: multiple R 2 0. Additionally, we can inspect the summary of the regression model as shown below to extarct additional information. Although Field, Miles, and Field suggest that the main effects of the predictors involved in the interaction should not be interpreted, they are interpreted here to illustrate how the results of a multiple linear regression can be reported.

The model fitting arrived at a final minimal model. During the model diagnostics, two outliers were detected and removed. Further diagnostics did not find other issues after the removal. The final minimal adequate regression model is based on 98 data points and performs highly significantly better than a minimal baseline model multiple R 2 :.

The final minimal adequate regression model reports attraction and status as significant main effects. This shows that men spend Whether men are attracted to women also correlates highly significantly and positively with the money they spend on women SE: 5. If men are not interested in women, they spend Furthermore, the final minimal adequate regression model reports a highly significant interaction between relationship status and attraction SE: 7. Logistic regression is a multivariate analysis technique that builds on and is very similar in terms of its implementation to linear regression but logistic regressions take dependent variables that represent nominal rather than numeric scaling Harrell Jr The difference requires that the linear regression must be modified in certain ways to avoid producing non-sensical outcomes.

The most fundamental difference between logistic and linear regressions is that logistic regression work on the probabilities of an outcome the likelihood , rather than the outcome itself. In addition, the likelihoods on which the logistic regression works must be logged logarithmized in order to avoid produce predictions that produce values greater than 1 instance occurs and 0 instance does not occur.

You can check this by logging the values from to 10 using the plogis function as shown below. If we visualize these logged values, we get an S-shaped curve which reflects the logistic function. To understand what this mean, we will use a very simple example. In this example, we want to see whether the height of men affect their likelihood of being in a relationship.

The data we use represents a data set consisting of two variables: height and relationship. In contrast to a linear regression, which predicts actual values, such as the frequencies of prepositions in a certain text, a logistic regression predicts probabilities of events for example, being in a relationship rather than actual values. The center panel shows the predictions of a logistic regression and we see that a logistic regression also has an intercept and a very steep slope but that the regression line also predicts values that are above 1 and below 0.

However, when we log the predicted values we these predicted values are transformed into probabilities with values between 0 and 1. And the logged regression line has a S-shape which reflects the logistic function.

Furthermore, we can then find the optimal line the line with the lowest residual deviance by comparing the sum of residuals - just as we did for a simple linear model and that way, we find the regression line for a logistic regression.

To exemplify how to implement a logistic regression in R see Agresti ; Agresti and Kateri for very good and thorough introductions to this topic], we will analyze the use of the discourse particle eh in New Zealand English and test which factors correlate with its occurrence.

The data set represents speech units in a corpus that were coded for the speaker who uttered a given speech unit, the gender, ethnicity, and age of that speaker and whether or not the speech unit contained an eh. To begin with, we clean the current work space, set option, install and activate relevant packages, load customized functions, and load the example data set. The summary of the data show that the data set contains 25, observations of five variables.

The variable ID contains strings that represent a combination file and speaker of a speech unit. The second variable represents the gender, the third the age, and the fourth the ethnicity of speakers.

The fifth variable represents whether or not a speech unit contained the discourse particle eh. Next, we factorize the variables in our data set. In other words, we specify that the strings represent variable levels and define new reference levels because as a default R will use the variable level which first occurs in alphabet ordering as the reference level for each variable, we redefine the variable levels for Age and Ethnicity.

After preparing the data, we will now plot the data to get an overview of potential relationships between variables. With respect to main effects, the Figure above indicates that men use eh more frequently than women, that young speakers use it more frequently compared with old speakers, and that speakers that are descendants of European settlers Pakeha use eh more frequently compared with Maori the native inhabitants of New Zealand.

The plots in the lower panels do not indicate significant interactions between use of eh and the Age, Gender, and Ethnicity of speakers. In a next step, we will start building the logistic regression model. As a first step, we need to define contrasts and use the datadist function to store aspects of our variables that can be accessed later when plotting and summarizing the model.

Contrasts define what and how variable levels should be compared and therefore influences how the results of the regression analysis are presented.

In this case, we use treatment contrasts which are in-built. Treatment contrasts mean that we assess the significance of levels of a predictor against a baseline which is the reference level of a predictor.

Field, Miles, and Field , —27 and Gries provide very good and accessible explanations of contrasts and how to manually define contrasts if you would like to know more. Next, we generate a minimal model that predicts the use of eh solely based on the intercept.

We will now start with the model fitting procedure. In the present case, we will use a manual step-wise step-up procedure during which predictors are added to the model if they significantly improve the model fit. In addition, we will perform diagnostics as we fit the model at each step of the model fitting process rather than after the fitting.

We will test two things in particular: whether the data has incomplete information or complete separation and if the model suffers from multi- collinearity. Incomplete information or complete separation means that the data does not contain all combinations of the predictor or the dependent variable. This is important because if the data does not contain cases of all combinations, the model will assume that it has found a perfect predictor. In such cases the model overestimates the effect of that that predictor and the results of that model are no longer reliable.

For example, if eh was only used by young speakers in the data, the model would jump on that fact and say Ha! Multicollinearity means that predictors correlate and have shared variance. This means that whichever predictor is included first will take all the variance that it can explain and the remaining part of the variable that is shared will not be attributed to the other predictor. This may lead to reporting that a factor is not significant because all of the variance it can explain is already accounted for.

However, if the other predictor were included first, then the original predictor would be returned as insignificant. This means that- depending on the order in which predictors are added - the results of the regression can differ dramatically and the model is therefore not reliable. Multicollinearity is actually a very common problem and there are various ways to deal with it but it cannot be ignored at least in regression analyses.

As the data does not contain incomplete information, the vif values are below 3, and adding Age has significantly improved the model fit the p-value of the ANOVA is lower than. We therefore proceed with Age included. We continue by adding Gender. If this were the case - if adding Gender would cause Age to become insignificant - then we could change the ordering in which we include predictors into our model. Again, including Gender significantly improves model fit and the data does not contain incomplete information or complete separation.

Also, including Gender does not affect the significance of Age. Now, we include Ethnicity. Since adding Ethnicity does not significantly improve the model fit, we do not need to test if its inclusion affects the significance of other predictors.

We continue without Ethnicity and include the interaction between Age and Gender. The interaction between Age and Gender is not significant which means that men and women do not behave differently with respect to their use of EH as they age.

Also, the data does not contain incomplete information and the model does not suffer from multicollinerity - the predictors are not collinear. We can now include if there is a significant interaction between Age and Ethnicity. Again, no incomplete information or multicollinearity and no significant interaction. Now, we test if there exists a significant interaction between Gender and Ethnicity. As the interaction between Gender and Ethnicity is not significant, we continue without it.

In a final step, we include the three-way interaction between Age , Gender , and Ethnicity. We have found our final minimal adequate model because the 3-way interaction is also insignificant.

As we have now arrived at the final minimal adequate model m2. After fitting the model, we validate the model to avoid arriving at a final minimal model that is overfitted to the data at hand. To validate a model, you can apply the validate function and apply it to a saturated model.

The output of the validate function shows how often predictors are retained if the sample is re-selected with the same size but with placing back drawn data points.

The execution of the function requires some patience as it is rather computationally expensive and it is, therefore, commented out below. The validate function shows that retaining two predictors Age and Gender is the best option and thereby confirms our final minimal adequate model as the best minimal model. In addition, we check whether we need to include a penalty for data points because they have too strong of an impact of the model fit.

To see whether a penalty is warranted, we apply the pentrace function to the final minimal adequate model. The values are so similar that a penalty is unnecessary. In a next step, we rename the final models. Now, we calculate a Model Likelihood Ratio Test to check if the final model performs significantly better than the initial minimal base-line model.

The result of this test is provided as a default if we call a summary of the lrm object. The p-value is lower than. Another way to extract the model likelihood test statistics is to use an ANOVA to compare the final minimal adequate model to the minimal base-line model. A handier way to get thses statistics is by performing an ANOVA on the final minimal model which, if used this way, is identical to a Model Likelihood Ratio test. In a next step, we calculate pseudo-R 2 values which represent the amount of residual variance that is explained by the final minimal adequate model.

We cannot use the ordinary R 2 because the model works on the logged probabilities rather than the values of the dependent variable. The low pseudo-R 2 values show that our model has very low explanatory power. For instance, the value of Hosmer and Lemeshow R 2 0. In essence, all the pseudo-R 2 values are measures of how substantive the model is how much better it is compared to a baseline model. Next, we extract the confidence intervals for the coefficients of the model. Despite having low explanatory and predictive power, the age of speakers and their gender are significant as the confidence intervals of the coefficients do not overlap with 0.

In a next step, we compute odds ratios and their confidence intervals. Odds Ratios represent a common measure of effect size and can be used to compare effect sizes across models. Odds ratios rang between 0 and infinity. Values of 1 indicate that there is no effect. The further away the values are from 1, the stronger the effect. If the values are lower than 1, then the variable level correlates negatively with the occurrence of the outcome the probability decreases while values above 1 indicate a positive correlation and show that the variable level causes an increase in the probability of the outcome the occurrence of EH.

The odds ratios confirm that older speakers use eh significantly less often compared with younger speakers and that women use eh less frequently than men as the confidence intervals of the odds rations do not overlap with 1. In a next step, we calculate the prediction accuracy of the model. In order to calculate the prediction accuracy of the model, we generate a variable called Prediction that contains the predictions of pour model and which we add to the data.

Then, we use the confusionMatrix function from the caret package Kuhn to extract the prediction accuracy. We can see that out model has never predicted the use of eh which is common when dealing with rare phenomena.

This is expected as the event s so rare that the probability of it not occurring substantively outweighs the probability of it occurring.

We are now in a position to perform model diagnostics and test if the model violates distributional requirements. In a first step, we test for the existence of multicollinearity.

Multicollinearity means that predictors in a model can be predicted by other predictors in the model this means that they share variance with other predictors. If this is the case, the model results are unreliable because the presence of absence of one predictor has substantive effects on at least one other predictor. To check whether the final minimal model contains predictors that correlate with each other, we extract variance inflation factors VIF.

Gries shows that a VIF of 10 means that that predictor is explainable predictable from the other predictors in the model with an R 2 of.

Indeed, predictors with VIF values greater than 4 are usually already problematic but, for large data sets, even VIFs greater than 2 can lead inflated standard errors Jaeger Also, VIFs of 2. See Gries or the excursion below for a more elaborate explanation.

During the workshop on mixed-effects modeling, we talked about multi- collinearity and someone asked if collinearity reflected shared variance what I thought or predictability of variables what the other person thought. Both answers are correct! We will see below why…. Multi- collinearity reflects the predictability of predictors based on the values of other predictors!

To test this, I generate a data set with 4 independent variables a , b , c , and d as well as two potential response variables r1 which is random and r2 where the first 50 data points are the same as in r1 but for the second 50 data points I have added a value of 50 to the data points 51 to from r1.

This means that the predictors a and d should both strongly correlate with r2. Now, I fit a first model. As the response is random, we do not expect any of the predictors to have a significant effect and we expect the R 2 to be rather low. We now check for multi- collinearity using the vif function from the rms package Harrell Jr Variables a and d should have high variance inflation factor values vif-values because they overlap very much! We now fit a second model to the response which has higher values for the latter part of the response.

Both a and d strongly correlate with the response. But because a and d are collinear, d should not be reported as being significant by the model. The R 2 of the model should be rather high given the correlation between the response r2 and a and d. The vif-values are identical which shows that what matters is if the variables are predictable. To understand how we arrive at vif-values, we inspect the model matrix. We now fit a linear model in which we predict d from the other predictors in the model matrix.

The R 2 shows that the values of d are explained to Now, we can write a function taken from Gries that converts this R 2 value. The function outputs the vif-value of d. In order to detect potential outliers, we will calculate diagnostic parameters and add these to our data set. In a next step, we use these diagnostic parameters to check if there are data points which should be removed as they unduly affect the model fit.

We now check whether the sample size is sufficient for our analysis Green According to rule of thumb provided in Green , the sample size is sufficient for our analysis. However, this statistic never reaches its theoretical maximum of 1. C is an index of concordance between the predicted probability and the observed response. When C takes the value 0. A value above 0.

Changes in AIC can serve as a measure of whether the inclusion of a variable leads to a significant increase in the amount of variance that is explained by the model.

These information criteria help you to decide. Within this model:. Ordinal regression is very similar to multiple linear regression but takes an ordinal dependent variable Agresti For this reason, ordinal regression is one of the key methods in analysing Likert data. And we will also factorize Internal and Exchange to make it easier to interpret the output later on.

Now that the dependent variable is releveled, we check the distribution of the variable levels by tabulating the data. To get a better understanding of the data we create frequency tables across variables rather than viewing the variables in isolation.

We also check the mean and standard deviation of the final score as final score is a numeric variable and cannot be tabulated unless we convert it to a factor. The lowest score is 1.

Finally, we inspect the distributions graphically. We see that we have only few students that have taken part in an exchange program and there are also only few internal students overall. With respect to recommendations, only few students are considered to very likely succeed in the program. We can now start with the modeling by using the polr function.

To make things easier for us, we will only consider the main effects here as this tutorial only aims to how to implement an ordinal regression but not how it should be done in a proper study - then, the model fitting and diagnostic procedures would have to be performed accurately, of course.

The results show that having studied here at this school increases the chances of receiving a positive recommendation but that having been on an exchange has a negative but insignificant effect on the recommendation. The final score also correlates positively with a positive recommendation but not as much as having studied here. As the regression report does not provide p-values, we have to calculate them separately after having calculated them, we add them to the coefficient table.

As predicted, Exchange does not have a significant effect but FinalScore and Internal both correlate significantly with the likelihood of receiving a positive recommendation. The odds ratios show that internal students are 2.

The effect of an exchange is slightly negative but, as we have seen above, not significant. This section is based on this tutorials on how to perform a Poisson regression in R.

NOTE Poisson regressions are used to analyze data where the dependent variable represents counts. This applied particularly to counts that are based on observations of something that is measured in set intervals. For instances the number of pauses in two-minute-long conversations. Poisson regressions are particularly appealing when dealing with rare events, i.

In such cases, normal linear regressions do not work because the instances that do occur are automatically considered outliers. Therefore, it is useful to check if the data conform to a Poisson distribution.

However, the tricky thing about Poisson regressions is that the data has to conform to the Poisson distribution which is, according to my experience, rarely the case, unfortunately. The Gaussian Normal Distribution is very flexible because it is defined by two parameters, the mean mu, i. This allows the normal distribution to take very different shapes for example, very high and slim compressed or very wide and flat. In contrast, the Poisson is defined by only one parameter lambda, i.

This is much trickier for natural data as this means that the Poisson distribution is very rigid.