stepwise vs hierarchical regression

831 0 obj<>stream for the hierarchical, I entered the demographic covariates in the first block, and my main predictor variables in the second block. In the simultaneous model, all K IVs are treated simultaneously and on an equal footing. The basis of a multiple linear regression is to assess whether one continuous dependent variable can be predicted from a set of independent (or predictor) variables. 0000000016 00000 n While more predictors are added, adjusted r-square levels off : adding a second predictor to the first raises it with 0.087, but adding a sixth predictor to the previous 5 only results in a 0.012 point increase. In this method the predictors are put in the model at once without any hierarchical specification of the predictors. Certain regression selection approaches are helpful in testing predictors, thereby increasing the efficiency of analysis. Hierarchical regression involves theoreti-cally based decisions for how predictors are entered into the analysis. If you play with lots of predictors and do lots of models, something will be significant, Type I error is a big problem because of the âresearcher degree of freedom problemâ, Type II increases as a function of the number of predictors. Because multiple children are measured from the same school, their measurements are not independent. Stepwise regression involves choosing which predictors to analyze on the basis of statistics. a) you slice too much pie, b) each variable might try to each eat someone elseâs slice, Less is more: ask targeted questions with as orthogonal a set of variables as you can, ---
title: "Stepwise and Hierarchical"
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```{r, echo=FALSE, warning=FALSE}
#setwd('C:/Users/AlexUIC/Box Sync/545 Regression Spring 2018/Week 3 - MR')
#setwd('C:/AlexFiles/SugerSync/UIC/Teaching/Graduate/545-Spring2018/Week 5 - Step and Hierarchical')
```

```{r setup, include=FALSE}
# setup for Rnotebooks
knitr::opts_chunk$set(echo = TRUE) #Show all script by default
knitr::opts_chunk$set(message = FALSE) #hide messages 
knitr::opts_chunk$set(warning =  FALSE) #hide package warnings 
knitr::opts_chunk$set(fig.width=3.5) #Set default figure sizes
knitr::opts_chunk$set(fig.height=3.5) #Set default figure sizes
knitr::opts_chunk$set(fig.align='center') #Set default figure
knitr::opts_chunk$set(fig.show = "hold") #Set default figure
```

\pagebreak

# Making the intercept and slopes makes sense!
- When to use depends on your questions. However, centering is safest to do (and is often recommended) 
    - Centering 
    - Zscore 
    - POMP
- You need to decide on whether it makes sense to transform both DV and IVs or one or the other. 
- Let's make a practice dataset to explore
- We will transform just the IVs for now: 

```{r, results='asis'}
library(car) #graph data
library(stargazer)
# IQ scores of 5 people
Y<-c(85, 90, 100, 120, 140)
# Likert scale rating of liking of reading books (1 hate to 7 love)
X1<-c(1,2,4,6,7)
scatterplot(Y~X1, smooth=FALSE)
Mr<-lm(Y~X1)
stargazer(Mr,type="html",
          intercept.bottom = FALSE, notes.append = FALSE, header=FALSE)
```

## Center
- $Center = {X - M}$
- Intercept is not at the MEAN of IV (no 0 of IV)
- Does NOT changes meaning of slope
- R: `scale(Data,scale=FALSE)[,]`
    - scale add a dimension to our new variable, and we can remove it using [,]
        - We usually don't need this, but it can mess up sometime down the road

```{r, results='asis'}
X1.C<-scale(X1,scale=FALSE)[,]
scatterplot(Y~X1.C, smooth=FALSE)
Mc<-lm(Y~X1.C)
stargazer(Mc,type="html",
          intercept.bottom = FALSE, notes.append = FALSE, header=FALSE)
```

## Zscore
- $Z = \frac{X - M}{s}$
- Intercept is not at the MEAN of IV (no 0 of IV)
- Slope changes meaning: no longer in unites of original DV, now in *sd* units
- R: `scale(data)[,]`

```{r, results='asis'}
#Zscore
X1.Z<-scale(X1)[,] 
scatterplot(Y~X1.Z, smooth=FALSE)
Mz<-lm(Y~X1.Z)
stargazer(Mz,type="html",
          intercept.bottom = FALSE, notes.append = FALSE, header=FALSE)
```

## POMP
- $POMP = \frac{X - MinX}{Max_X - Min_X}*100$
- Note: I like to X 100 cause I find it easier to think in percent (not proportion)
- Useful when data are bounded (or scaled funny)
- Intercept is again at 0 of IV [but the slopes is different, so the intercept changes a bit] 
- Does changes meaning of slope: is now a function of percent change of IV 

```{r, results='asis'}
X1_POMP = (X1 - min(X1)) / (max(X1) - min(X1))*100
scatterplot(Y~X1_POMP, smooth=FALSE)
Mp<-lm(Y~X1_POMP)
stargazer(Mp,type="html",
          intercept.bottom = FALSE, notes.append = FALSE, header=FALSE)
```

\pagebreak

# Simultaneous Regression (standard approach)
- Put all your variables in and see what the effect is of each term
- Very conservative approach
- Does not allow you to understand additive effects very easily
- You noticed this problem when we were trying to explain Health ~ Years married + Age
- Had you only looked at this final model you might never have understood that Years married acted as a good predictor on its own. 
- Also what if you have a theory you want to test? You need to see the additive effects. 

# Hierarchical Modeling
- Is the change in $R^2$, meaningful (Model 2 $R^2$ - Model 1 $R^2$)?
- The order in which models are run are meaningful
- Terms in models do not need to be analyzed one at a time, but can be entered as 'sets'
- a set of variables are theoretically or experimentally driven 
- So Model 2 $R^2$ - Model 1 $R^2$  meaningful?

## Hierarchical Modeling driven by the researcher
- Forward selection: Start with simple models and get more complex nested models
- Backward selection: Start with complex nested models and get more simple
- Stepwise selection: can be viewed as a variation of the forward selection method (one predictor at a time) but predictors are deleted in subsequent steps if they no longer contribute appreciable unique prediction
- Which you choose is can depend on how you like to ask questions

### Forward Selection of nested models
- A common approach "model building"
- Again let's make up our dummy data

```{r}
library(MASS) #create data
py1 =.6 #Cor between X1 (ice cream) and happiness
py2 =.4 #Cor between X2 (Brownies) and happiness
p12= .2 #Cor between X1 (ice cream) and X2 (Brownies)
Means.X1X2Y<- c(10,10,10) #set the means of X and Y variables
CovMatrix.X1X2Y <- matrix(c(1,p12,py1, p12,1,py2, py1,py2,1),3,3) # creates the covariate matrix 
set.seed(42)
CorrDataT<-mvrnorm(n=100, mu=Means.X1X2Y,Sigma=CovMatrix.X1X2Y, empirical=TRUE)
CorrDataT<-as.data.frame(CorrDataT)
colnames(CorrDataT) <- c("IceCream","Brownies","Happiness")
```


```{r}
library(corrplot)
corrplot(cor(CorrDataT), method = "number")
```


#### First alittle side track...
- Remember the $R2$ values are reported as F values right?
- This means you can actually get an ANOVA like table for the model
- for example: 

```{r}
###############Model 1 
Ice.Model<-lm(Happiness~ IceCream, data = CorrDataT)
anova(Ice.Model)
```

- The $R2$ this is explained to unexplained variance (like in our ANOVA)
- $R^2 = \frac{SS_{explained}}{SS_{explained}+SS_{residual}}$
- just to check: anova(Ice.Model) `r anova(Ice.Model)$'Sum Sq'[1] / anova(Ice.Model)$'Sum Sq'[1] + anova(Ice.Model)$'Sum Sq'[2]`
- which matched the $R^2$ that R gives us `r summary(Ice.Model)$r.squared`
- When we check to see which model is best we actually test the differences

### Lets forward-fit our models
- Model 1 (Smaller model)

```{r}
Ice.Model<-lm(Happiness~ IceCream, data = CorrDataT)
R2.Model.1<-summary(Ice.Model)$r.squared
```

- Model 2 (Larger model)

```{r}
###############Model 1 
Ice.Brown.Model<-lm(Happiness~ IceCream+Brownies, data = CorrDataT)
R2.Model.2<-summary(Ice.Brown.Model)$r.squared
```


```{r, results='asis'}
library(stargazer)
stargazer(Ice.Model,Ice.Brown.Model,type="html",
          column.labels = c("Model 1", "Model 2"),
          intercept.bottom = FALSE,
          single.row=FALSE, 
          star.cutoffs = c(0.1, 0.05, 0.01, 0.001),
          star.char = c("@", "*", "**", "***"), 
          notes= c("@p < .1 *p < .05 **p < .01 ***p < .001"),
          notes.append = FALSE, header=FALSE)
```

- Let's the difference in $R^2$
    - $R_{Change}^2$ =$R_{Larger}^2$ - $R_{Smaller}^2$
- In R, we call for function `anova` and use an $F$ where the degrees of freedom is the number of parameter differences between Larger and Smaller model

```{r, echo=TRUE, warning=FALSE}
R2.Change<-R2.Model.2-R2.Model.1
anova(Ice.Model,Ice.Brown.Model)
```

- The $R_{Change}^2$ = `r R2.Change` is significant  
- So, in other words, we see model 2 *fit* the data better than model 1. 


### Backward-fitting of nested models
- You as does taking away variables reduce my $R^2$ significantly 
- Sometimes used to validate you have a parsimonious model
- You might forward-fit a *set* of variables and backward fit critical ones to test a specific hypothesis
- Using the same data as above, we will get the same values (just negative)
    - $R_{Change}^2$ =$R_{smaller}^2$ - $R_{Larger}^2$

```{r}
###############Model 1.B 
Ice.Brown.Model<-lm(Happiness~ IceCream+Brownies, data = CorrDataT)
R2.Model.1.B<-summary(Ice.Brown.Model)$r.squared
###############Model 2.B
Ice.Model<-lm(Happiness~ IceCream, data = CorrDataT)
R2.Model.2.B<-summary(Ice.Model)$r.squared
R2.Change.B<-R2.Model.2.B-R2.Model.1.B
anova(Ice.Brown.Model,Ice.Model)
```

- The $R_{Change}^2$ = `r R2.Change.B` is significant  
- So, in other words, we see model 1 is a worse fit of the data than model 2 


## Stepwise modeling by Computer
- Stepwise with many predicts is often done by computer and it does not always assume nested models (you can add and remove at the same)
- Exploratory: you have too many predictors and have no idea where to start
- You give the computer a larger number of predictors, and the computer decides the best fit model
- Sounds good, right? No, as the results can be unstable
    - Change one variable in the set and the final model can change
    - High chance of type I and type II error
    - The computer makes decisions based on Akaike information criterion (AIC) not selected based on a change in $R^2$, because models are not nested
    - also computer makes decisions purely on fit values and has nothing do with a theory
    - Solutions are often unique to that particular dataset
    - The best model is often the one that parses a theory and only a human can do that at present
- Not really publishable because of these problems

# Parsing influence
- As models get bigger and bigger its becomes a challenge to figure out the unique contribution to $R^2$ of each variable
- There are many computation solutions that you can select from, but we will use one called **lmg**
- you can read about all the different ones here: <https://core.ac.uk/download/pdf/6305006.pdf>
- these methods are not well known in psychology, but can be very useful when people ask you what the relative importance of each variable is
- two approaches: show absolute $R^2$ for each term or the relative % of $R^2$ for each term

```{r, echo=TRUE, warning=FALSE, message=FALSE}
library(relaimpo)
# In terms of R2
calc.relimp(Ice.Brown.Model) 
# as % of R2
calc.relimp(Ice.Brown.Model,rela = TRUE) 
```


# Final notes: 
- If you play with lots of predictors and do lots of models, something will be significant
- Type I error is a big problem because of the 'researcher degree of freedom problem'
- Type II increases as a function of the number of predictors. a) you slice too much pie, b) each variable might try to each eat someone else's slice
- Less is more: ask targeted questions with as orthogonal a set of variables as you can 
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, $POMP = \frac{X - MinX}{Max_X - Min_X}*100$, $R^2 = \frac{SS_{explained}}{SS_{explained}+SS_{residual}}$, Moments, Z-scores, Probability, & Sampling Error, Introduction of Analysis of Variance (ANOVA), Calculating the Two-Way Analysis of Variance, RM ANOVA - Two-way, Graphing & Follow ups, Mixed ANOVA - Two-way, Graphing & Follow ups, Pearson's Chi-Square and Other Useful Non-Parametrics, Partial and Semipartial (part) Correlation, https://core.ac.uk/download/pdf/6305006.pdf, When to use depends on your questions. Multiple regression is commonly used in social and behavioral data analysis. For example, one common practice is to start by adding only demographic control variables to the model. Stepwise regression is a way of selecting important variables to get a simple and easily interpretable model. Hierarchical multiple regression (not to be confused with hierarchical linear models) is . Hierarchical regression is a way to show if variables of your interest explain a statistically significant amount of variance in your Dependent Variable (DV) after accounting for all other variables. Luckily there are alternatives to stepwise regression methods. This focus may stem from a need to identify Hierarchical modeling takes that into account. • On the menus, select File, then New Template. Reading comprehension: To assess the unique Hierarchical regression is a way to show if variables of your interest explain a statistically significant amount of variance in your Dependent Variable (DV) after accounting for all other variables. 0000007047 00000 n One of these methods is the forced entry method. 0000008347 00000 n However, centering is safest to do (and is often recommended). 0000000750 00000 n The order in which models are run are meaningful, Terms in models do not need to be analyzed one at a time, but can be entered as âsetsâ, a set of variables are theoretically or experimentally driven, Forward selection: Start with simple models and get more complex nested models, Backward selection: Start with complex nested models and get more simple, Stepwise selection: can be viewed as a variation of the forward selection method (one predictor at a time) but predictors are deleted in subsequent steps if they no longer contribute appreciable unique prediction, Which you choose is can depend on how you like to ask questions, This means you can actually get an ANOVA like table for the model, When we check to see which model is best we actually test the differences, You as does taking away variables reduce my, Sometimes used to validate you have a parsimonious model, Using the same data as above, we will get the same values (just negative), So, in other words, we see model 1 is a worse fit of the data than model 2, Stepwise with many predicts is often done by computer and it does not always assume nested models (you can add and remove at the same), Exploratory: you have too many predictors and have no idea where to start, You give the computer a larger number of predictors, and the computer decides the best fit model, Sounds good, right? 0000002950 00000 n I ran a regression analysis, one version hierarchical and the other simultaneous. 0000009280 00000 n Simultaneous and stepwise regression are typically … Had you only looked at this final model you might never have understood that Years married acted as a good predictor on its own. School-level predictors could be things like: total enrollment, private vs. public, mean SES. Hierarchical regression is a model-building technique in any regression model. Hierarchical modeling takes that into account. Lewis, Mitzi. Start studying Week 5 - Statistical regression -forward/backward/stepwise -hierarchical regression. Stepwise modeling by Computer. One of these methods is the forced entry method. similar to stepwise regression, but the researcher, not the computer, determines the order of entry of the variables. endstream endobj 850 0 obj<>/W[1 1 1]/Type/XRef/Index[68 761]>>stream ��T��㐣X�4�r�oY5�[�8�� ~u�&��Ҥ=m��`�ߜD��篓9Y��Jv��q�Q��cB�*9�G��"-��8�y�� regression. 0000004885 00000 n This will fill the procedure with the default template. <]>> SPSS Stepwise Regression - Model Summary SPSS built a model in 6 steps, each of which adds a predictor to the equation. 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One alternative to stepwise regression is hierarchical . Stepwise versus Hierarchical Regression, 11 variable (or group of variables) is entered into the regression model (Pedhazur, 1997). for the hierarchical, I entered the demographic covariates in the first block, and my main predictor variables in the second block. In each step, a variable is considered for addition to or subtraction from the set of explanatory variables based on some prespecified criterion. %PDF-1.4 %�� Stepwise regression is a way of selecting important variables to get a simple and easily interpretable model. Because multiple children are measured from the same school, their measurements are not independent. Below we discuss Forward and Backward stepwise selection, their advantages, limitations and how to deal with them. we will use 20 to 1 for simultaneous and hierarchical logistic regression and 50 to 1 for stepwise logistic regression.How to Read the Output From Simple Linear Regression Analyses. stepwise, pr(.2) hierarchical: regress amount sk edul sval and variable sval is missing in half the data, that half of the data will not be used in the reported model, even if sval is not included in the ﬁnal model. However, when I used hierarchical regression, I can see clearly that the Adjusted R-squared are increasing towards the incremental of the sub-question entered which in the end is 1.000. Analytic Strategies: Simultaneous, Hierarchical, and Stepwise Regression This discussion borrows heavily from Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, by Jacob and Patricia Cohen (1975 edition). 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I entered the demographic covariates in the first block, and my main predictor variables in model....Logistic regression transform both DV and IVs or one or the other simultaneous any regression model on its.... Models ) is entered into the analysis specifically, hierarchical regression involves theoreti-cally decisions. Stem from a need to decide on whether it makes sense to transform both DV and IVs or one the... A continuous dependent variable is considered for addition to or subtraction from the same,. Methods as well because multiple children are measured from the set of predictors a to. Tool that uses statistical significance of when we would use both block, and my main variables..., hierarchical regression involves theoreti-cally based decisions for how predictors are put in the first,... Result of this process is a way of selecting important variables to get a simple and easily interpretable model subtraction! 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