No interaction

For the sake of experiment we can compare two models based solely on the number of books at home and presence of either personal or shared tablet. Since models are not nested, we have to use AIC instead of ANOVA.

## [1] 52203.08

## [1] 52257.89

AIC of the first model appears to be lower 52203.08 > 52257.89 and we conclude that it is better. Additionally, it explains slightly more variance in the outcome variable (0.02787 > 0.01557) while both models are better than no model at all, judjing by the p-value being less than 0.05.

We then proceed to include control variables in the equation and test whether the addition of two tablet predictor variables improves the model that is focused on the number of books at home only.

## Analysis of Variance Table
## 
## Model 1: math ~ books + gender + country + edmother + edfather
## Model 2: math ~ books + selftab + sharedtab + gender + country + edmother + 
##     edfather
##   Res.Df      RSS Df Sum of Sq      F    Pr(>F)    
## 1   4496 25361460                                  
## 2   4494 25028212  2    333248 29.919 1.237e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Since P-value is small (1.237e-13) we conclude that the additon of two new predictors did improve the model. The final model includes both the number of books and presence of shared or personal tablet as well as control variables.

The final model appears to be better than no model at all (p-value: < 2.2e-16) and explains 11% of the variance in the outcome variable. Three factor levels of the number of books at home (base level excluded) were coded as statistically significant, while none of the parental education variables were. Gender and both tablet variables were found to be statistically significant, while country of origin wasn’t.

While the regression equation would look a bit confusing with that many variables, the interpretation can be presented as:

• When all variables are equal to 0 (meaning that respondent is a girl, has a personal tablet and a shared tablet, was born in Russia, has less than 10 books at home and both parents obtained a “some primary or lower secondary education” education) the predicted math score is equal to 502.6493.

• The increase of the predicted math score in comparison to the base category of books (less than 10 books at home) is getting bigger as we move to levels indicating larger numbers of books at home, however, an increase associated with “More than 200” level is smaller than that of “101–200 books” (18.6710 < 23.6901).

• If respondent doesn’t have a personal tablet, the predicted math score is higher by 14.1105.

• If respondent doesn’t have a shared tablet, the predicted math score is lower by 18.6497.

• If respondent is a boy, the predicted math score is higher by 8.3003.

• If respondent wasn’t born in Russia, the predicted math score is lower by 0.3096.

• Starting from the “Post-secondary, non-tertiary” change in the education level of mother and father is associated with an increase in the predicted math score. Increase attributed to the “Short-cycle tertiary” is lower than that of “Post-secondary, non-tertiary” and “Bachelor’s or equivalent” in both cases. While in case of mother’s education by ascribing to the “Don’t know” category” increases the predicted math score (7.9611), for father’s education it actually decreases it (-3.3976).

The findings are consistent with the hypothesis presented above: presence of a personal tablet is associated with a decrease in predicted math achievement score, presence of a shared tablet is associated with an increase in predicted math achievement score, increase in the number of books at home is associated with an increase in the predicted math achievement score.

Interaction

We now proceed to add a new variable as an interaction term, but before doing so, we need to take a closer look at it. The variable we want to include it’s respondents own educational aspirations, namely, the level of education he or she is planning to obtain. The reason behind the addition of this variable is the following: we add personal educational aspirations of a student as an interaction term to the varible indicating the number of books at home. We might hypothesize that the effect of the number of books at home on the math achievement will be different for different levels of educational aspiration: high educational aspirations (and willingness to work to achieve one’s goal) might compensate for the lack of either abilities or opportunities priveded by teachers at school that a small number of books at home led to.

The largest share of respondents stated that they want to obtain a Bachelor’s degree (36.7%), followed by the 19.8% of respondents claiming that they want to obtain a postgraduate degree. The general trend with the median math achievement was upward: the higher were the educational aspirations of respondents, the higher were their median math achievement score. That is also demonstrated in the descriptive table below.

Finally, we can perform ANOVA to check if the difference in math scores for these groups is statistically significant.

##                  Df   Sum Sq Mean Sq F value Pr(>F)    
## data1$futureed    4  4809646 1202412   230.2 <2e-16 ***
## Residuals      4512 23571760    5224                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Since P-value < .001 (<2e-16), we conlude that the difference in math scores across different levels of educational aspiration is statistically significant.

The interaction effect is significant on several levels: 101–200 books at home and aspired to finish Bachelor’s or equivalent; 101–200 books at home and aspired to obtain Postgraduate degree; more than 200 books at home and aspired to obtain Postgraduate degree. Three out of four levels of the variable educational aspiration (base level excluded) were found to be statistically significant, while the variable indicating the number of books at home lost it statistical significance. The model is still better than no model at all and explains around 20% of the variance in the outcome variable.

We can also take a look at the interaction plot presented above. Predcited math scores of those with higher educational aspirations are always higher and the lines do not overlap. Predicted math score drops at the “More than 200 books at home” level for three out of five educational aspiration types: Upper secondary, Post-secondary but non-tertiary and Bachelor’s. For those aspired to obrain Postgraduate degree or only finish lower secondary education the predicted math score increases at that level of number of books compared to the previous one. Another pecularity in case of the “Finish lower secondary” aspiration level is that the predicted math score drops at the level of 101-200 books at home in comparison to the previous one unlike it happens in all four remaining aspiration levels. In general it can be seen that the increase in predicted math score on each level compared to the previous one is more steep for such high aspiration level as Postgraduate degee and it is the only level for which the predicted math achievement always increases.

Now we want to compare the model with and without interaction to check which one if better.

## Analysis of Variance Table
## 
## Model 1: math ~ books + selftab + sharedtab + gender + country + edmother + 
##     edfather
## Model 2: math ~ books * futureed + selftab + sharedtab + gender + country + 
##     edmother + edfather
##   Res.Df      RSS Df Sum of Sq     F    Pr(>F)    
## 1   4494 25028212                                 
## 2   4474 22596800 20   2431412 24.07 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Since P-value is small (2.2e-16) we conclude that the additon of the interaction did improve the model and claim it to be our final one.

Model diagnostics

In model diagnostics, we can first check for outliers.

## No Studentized residuals with Bonferroni p < 0.05
## Largest |rstudent|:
##       rstudent unadjusted p-value Bonferroni p
## 4173 -3.576513         0.00035189           NA

## 1712 4173 
## 1615 3944

We observe that there is an outlier observation - 4173 (at the low left corner of the QQ Plot). We will have to check whether it is an influential outlier later on the diagnostic plots.

In order to check for multicollinearity we calculate variance inflation factors for all the variables.

##                        GVIF Df GVIF^(1/(2*Df))
## books          1.085381e+03  4        2.395785
## futureed       6.492610e+04  4        3.995328
## selftab        1.019190e+00  1        1.009549
## sharedtab      1.028435e+00  1        1.014118
## gender         1.053147e+00  1        1.026229
## country        1.018837e+00  1        1.009374
## edmother       4.139937e+00  7        1.106805
## edfather       3.571689e+00  7        1.095194
## books:futureed 3.172123e+07 16        1.715605

In order to make GVIF^(1/(2 X DF)) analogous to the standard VIF measure, we need to square it. As we observe some squared values of GVIF^(1/(2 X DF)) exceed the threshold of 5 (15.962644 and 5.739784) which is a problem and indicates multicillinearity. Meaning, some of our explanatory variables are highly linearly correlated and it might be hard to distinguish their individual effects on the outome variable.

We then proceed to check for heteroscedasticity of residuals.

## Non-constant Variance Score Test 
## Variance formula: ~ fitted.values 
## Chisquare = 18.10096, Df = 1, p = 2.095e-05

Unfortunately, p-value is statistically significant (2.095e-05<0.05) which is a sign of heteroscedasticity (the residuals don’t have a constant variance).

One of possible explanations for this is that, heteroscedasticity is quite a common phenomena in cross-sectional. Additionally, Russia is a relatively big country with quite different regions and while extremely high values of some variables can be observed in one area, the other area might have extremely low values of the same variables. We will also observe heteroscedasticity on the diagnostic plots in what follows.

• The horizontal line on the “Residuals vs Fitted” graph seems to be quite straight, which is good since here we are checking the linearity.

• On the second graph we observe the outliers we’ve already mentioned. Apart from those, the line is actually quite straight.

• “Scale-Location” graph is used to check the homogeneity of variance of the residuals and the ideal case would be a horizontal line with points equally spread around it - that is not our case. The line on our plot seems to have a slightly negative slope.

• There are no points on the Line of Cook’s distance which means there are no high influence outliers that should be deleted from the model.

No rotation, five factors

## Factor Analysis using method =  minres
## Call: fa(r = datanum, nfactors = 5, rotate = "none", cor = "mixed")
## Standardized loadings (pattern matrix) based upon correlation matrix
##            MR1   MR2   MR3   MR4   MR5   h2    u2 com
## selftab   0.56 -0.36 -0.09 -0.54  0.28 0.82 0.184 3.3
## sharedtab 0.24  0.54 -0.08  0.15  0.22 0.43 0.574 2.0
## desk      0.71 -0.38 -0.49  0.27 -0.18 1.00 0.005 2.9
## room      0.45 -0.31 -0.02  0.23  0.06 0.36 0.643 2.4
## internet  0.66  0.36 -0.19 -0.07  0.14 0.62 0.378 1.9
## phone     0.65  0.38 -0.12 -0.10 -0.13 0.60 0.395 1.8
## game      0.37 -0.06  0.35 -0.27 -0.18 0.37 0.629 3.3
## music     0.29  0.17  0.06  0.08 -0.17 0.15 0.849 2.7
## car       0.48  0.10  0.29  0.12  0.14 0.36 0.643 2.1
## flat      0.36 -0.23  0.41  0.34  0.27 0.53 0.468 4.4
## dish      0.54  0.01  0.38 -0.03 -0.30 0.52 0.477 2.4
## 
##                        MR1  MR2  MR3  MR4  MR5
## SS loadings           2.80 1.03 0.82 0.66 0.44
## Proportion Var        0.25 0.09 0.07 0.06 0.04
## Cumulative Var        0.25 0.35 0.42 0.48 0.52
## Proportion Explained  0.49 0.18 0.14 0.12 0.08
## Cumulative Proportion 0.49 0.66 0.81 0.92 1.00
## 
## Mean item complexity =  2.7
## Test of the hypothesis that 5 factors are sufficient.
## 
## df null model =  55  with the objective function =  2.41 with Chi Square =  10886.92
## df of  the model are 10  and the objective function was  0.03 
## 
## The root mean square of the residuals (RMSR) is  0.01 
## The df corrected root mean square of the residuals is  0.03 
## 
## The harmonic n.obs is  4517 with the empirical chi square  69.41  with prob <  5.8e-11 
## The total n.obs was  4517  with Likelihood Chi Square =  128.97  with prob <  7.6e-23 
## 
## Tucker Lewis Index of factoring reliability =  0.94
## RMSEA index =  0.051  and the 90 % confidence intervals are  0.044 0.059
## BIC =  44.81
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    MR1  MR2  MR3  MR4  MR5
## Correlation of (regression) scores with factors   0.95 0.87 0.86 0.85 0.74
## Multiple R square of scores with factors          0.90 0.76 0.74 0.73 0.54
## Minimum correlation of possible factor scores     0.81 0.53 0.48 0.46 0.09

Let’s first assess the model fit by the produced output.

• Cumulative proportion of variance explained by all factors is equal to 0.52 which is relatively high.
• ‘Proportion Var’ values are equal to 0.25, 0.09, 0.07, 0.06 and 0.04 with all of them being bigger 0.1.
• We want the values of ‘Proportion Explained’ to be more or less similar. However, we observe that they range from 0.49 to 0.08 which is far from being ideal. Values for MR2 (0.18), MR3 (0.14), and MR4 (0.12) are actually quite similar and the problem is simply in the first factor having that high of a value (0.49).
• The root mean square of the residuals (RMSR) is 0.01 which is good since we want it to be closer to zero and at least less than 0.05.
• RMSEA index is equal to 0.051 which is, in fact, we good (<.08 acceptable, <.05 excellent).
• Tucker Lewis index of factoring reliability is equal to 0.94 which also makes it acceptable (>.90 acceptable, >.95 excellent).

In general, it seems like the fit of the model is quite good apart from one of the factors having a too high ‘proportion explained’ value. However, if we look at the plot we observe that there are multiple problems present: two of the factors have only one variable loaded on them, while two have none. The majority of variables are loaded on the first factor which can also be seen in the output. It’s quite natural to have such a bad picture since we haven’t performed the rotation yet.

Varimax rotation, five factors

While keeping the number of factors to 5 we proceed to perform the varimax (orthogonal) rotation which assumes that factors are not correlated. We suppose that after the rotation the indicators will load high on one factor and one factor only - we will basically try to maximize each indicator’s largest factor loading and minimize others.

## Factor Analysis using method =  minres
## Call: fa(r = datanum, nfactors = 5, rotate = "varimax", cor = "mixed")
## Standardized loadings (pattern matrix) based upon correlation matrix
##             MR2   MR1   MR3   MR4   MR5   h2    u2 com
## selftab    0.06  0.22  0.15  0.85  0.09 0.82 0.184 1.2
## sharedtab  0.63 -0.09 -0.06 -0.11  0.10 0.43 0.574 1.2
## desk       0.16  0.96  0.10  0.16  0.07 1.00 0.005 1.1
## room       0.02  0.46  0.09  0.12  0.35 0.36 0.643 2.1
## internet   0.70  0.21  0.17  0.24  0.05 0.62 0.378 1.6
## phone      0.62  0.23  0.38  0.13 -0.07 0.60 0.395 2.1
## game       0.00 -0.03  0.55  0.24  0.11 0.37 0.629 1.5
## music      0.23  0.12  0.27 -0.10  0.04 0.15 0.849 2.7
## car        0.29  0.05  0.28  0.08  0.42 0.36 0.643 2.7
## flat      -0.01  0.12  0.13  0.03  0.71 0.53 0.468 1.1
## dish       0.12  0.14  0.67  0.04  0.21 0.52 0.477 1.4
## 
##                        MR2  MR1  MR3  MR4  MR5
## SS loadings           1.45 1.35 1.13 0.93 0.89
## Proportion Var        0.13 0.12 0.10 0.08 0.08
## Cumulative Var        0.13 0.25 0.36 0.44 0.52
## Proportion Explained  0.25 0.23 0.20 0.16 0.15
## Cumulative Proportion 0.25 0.49 0.68 0.85 1.00
## 
## Mean item complexity =  1.7
## Test of the hypothesis that 5 factors are sufficient.
## 
## df null model =  55  with the objective function =  2.41 with Chi Square =  10886.92
## df of  the model are 10  and the objective function was  0.03 
## 
## The root mean square of the residuals (RMSR) is  0.01 
## The df corrected root mean square of the residuals is  0.03 
## 
## The harmonic n.obs is  4517 with the empirical chi square  69.41  with prob <  5.8e-11 
## The total n.obs was  4517  with Likelihood Chi Square =  128.97  with prob <  7.6e-23 
## 
## Tucker Lewis Index of factoring reliability =  0.94
## RMSEA index =  0.051  and the 90 % confidence intervals are  0.044 0.059
## BIC =  44.81
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    MR2  MR1  MR3  MR4  MR5
## Correlation of (regression) scores with factors   0.85 0.99 0.78 0.88 0.77
## Multiple R square of scores with factors          0.72 0.98 0.60 0.78 0.59
## Minimum correlation of possible factor scores     0.44 0.96 0.20 0.56 0.19

We observe how the values of “Proportion explained” changed: while they used to range from 0.49 to 0.08, they now range from 0.15 to 0.25 which is a lot better (MR2 now has to biggest value). We also observe how the plot changed: now all factors have at least one variables loaded on them, but only one of the factors have a sufficient number of variables loading on it equal to three. Changes are also apparent for the factor loadings: for instance, “selftab” that was previously loaded on the MR1 factor is now mainly loaded on MR4 with 0.85 factor loading, “sharedtab” that was initially loaded on the MR2 factor with 0.54 is now loaded on it with 0.63. Changes in other varaibles are also presented in the output.

Since varimax (orthogonal) rotation assumes that factors are not correlated, the correlations are not seen in the output.

Oblimin rotation, five factors

## Factor Analysis using method =  minres
## Call: fa(r = datanum, nfactors = 5, rotate = "oblimin", cor = "mixed")
## Standardized loadings (pattern matrix) based upon correlation matrix
##             MR2   MR1   MR3   MR4   MR5   h2    u2 com
## selftab    0.02  0.01 -0.01  0.90  0.01 0.82 0.184 1.0
## sharedtab  0.69 -0.12 -0.14 -0.12  0.10 0.43 0.574 1.3
## desk       0.01  1.00 -0.01  0.01  0.00 1.00 0.005 1.0
## room      -0.05  0.43  0.01  0.07  0.32 0.36 0.643 1.9
## internet   0.68  0.11  0.03  0.19 -0.01 0.62 0.378 1.2
## phone      0.56  0.15  0.30  0.04 -0.16 0.60 0.395 1.9
## game      -0.07 -0.14  0.56  0.20  0.02 0.37 0.629 1.4
## music      0.19  0.11  0.27 -0.17 -0.01 0.15 0.849 2.9
## car        0.28 -0.05  0.21  0.05  0.38 0.36 0.643 2.5
## flat      -0.01  0.04  0.05  0.02  0.70 0.53 0.468 1.0
## dish       0.02  0.06  0.68 -0.06  0.11 0.52 0.477 1.1
## 
##                        MR2  MR1  MR3  MR4  MR5
## SS loadings           1.45 1.33 1.14 0.98 0.85
## Proportion Var        0.13 0.12 0.10 0.09 0.08
## Cumulative Var        0.13 0.25 0.36 0.45 0.52
## Proportion Explained  0.25 0.23 0.20 0.17 0.15
## Cumulative Proportion 0.25 0.48 0.68 0.85 1.00
## 
##  With factor correlations of 
##      MR2  MR1  MR3  MR4  MR5
## MR2 1.00 0.27 0.31 0.17 0.06
## MR1 0.27 1.00 0.27 0.40 0.17
## MR3 0.31 0.27 1.00 0.34 0.25
## MR4 0.17 0.40 0.34 1.00 0.12
## MR5 0.06 0.17 0.25 0.12 1.00
## 
## Mean item complexity =  1.6
## Test of the hypothesis that 5 factors are sufficient.
## 
## df null model =  55  with the objective function =  2.41 with Chi Square =  10886.92
## df of  the model are 10  and the objective function was  0.03 
## 
## The root mean square of the residuals (RMSR) is  0.01 
## The df corrected root mean square of the residuals is  0.03 
## 
## The harmonic n.obs is  4517 with the empirical chi square  69.41  with prob <  5.8e-11 
## The total n.obs was  4517  with Likelihood Chi Square =  128.97  with prob <  7.6e-23 
## 
## Tucker Lewis Index of factoring reliability =  0.94
## RMSEA index =  0.051  and the 90 % confidence intervals are  0.044 0.059
## BIC =  44.81
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    MR2  MR1  MR3  MR4  MR5
## Correlation of (regression) scores with factors   0.87 1.00 0.83 0.91 0.79
## Multiple R square of scores with factors          0.76 1.00 0.69 0.83 0.62
## Minimum correlation of possible factor scores     0.52 0.99 0.37 0.66 0.24

We then perform the “oblimin” rotation and immediately see the correlations between factors on the plot: MR1 and MR4 - 0.4, MR2 and MR3 - 0.3, MR3 and MR4 - 0.3. Since the results of the tests won’t change we have to look at the changes in SS loadings, proportion variance etc: however, the “Proportion Explained” values, in fact, almost didn’t change.

Since there is no significant improvement in comparison to the results of the varimax rotation, the second model appears to be preferable.

Varimax rotation, three factors

As we observed on the plots, the number of variables loaded on factors was not sufficient with only one factor having three variables loaded on it. In order to solve this problem we can reduce the number of factors. Since in case of the four factors some of them still have less than three variables loaded on them, we proceed to reduce the number of factors to three.

## Factor Analysis using method =  minres
## Call: fa(r = datanum, nfactors = 3, rotate = "varimax", cor = "mixed")
## Standardized loadings (pattern matrix) based upon correlation matrix
##             MR2   MR1   MR3   h2   u2 com
## selftab    0.13  0.40  0.25 0.24 0.76 1.9
## sharedtab  0.53 -0.10 -0.02 0.29 0.71 1.1
## desk       0.21  0.86  0.05 0.79 0.21 1.1
## room      -0.01  0.56  0.23 0.37 0.63 1.3
## internet   0.73  0.28  0.14 0.64 0.36 1.4
## phone      0.68  0.23  0.23 0.57 0.43 1.5
## game       0.07  0.07  0.54 0.30 0.70 1.1
## music      0.26  0.06  0.19 0.11 0.89 2.0
## car        0.27  0.13  0.46 0.30 0.70 1.8
## flat      -0.04  0.22  0.39 0.20 0.80 1.6
## dish       0.19  0.16  0.61 0.44 0.56 1.3
## 
##                        MR2  MR1  MR3
## SS loadings           1.52 1.47 1.26
## Proportion Var        0.14 0.13 0.11
## Cumulative Var        0.14 0.27 0.39
## Proportion Explained  0.36 0.34 0.30
## Cumulative Proportion 0.36 0.70 1.00
## 
## Mean item complexity =  1.5
## Test of the hypothesis that 3 factors are sufficient.
## 
## df null model =  55  with the objective function =  2.41 with Chi Square =  10886.92
## df of  the model are 25  and the objective function was  0.32 
## 
## The root mean square of the residuals (RMSR) is  0.05 
## The df corrected root mean square of the residuals is  0.08 
## 
## The harmonic n.obs is  4517 with the empirical chi square  1370.03  with prob <  9.6e-274 
## The total n.obs was  4517  with Likelihood Chi Square =  1421.89  with prob <  8.1e-285 
## 
## Tucker Lewis Index of factoring reliability =  0.716
## RMSEA index =  0.111  and the 90 % confidence intervals are  0.106 0.116
## BIC =  1211.5
## Fit based upon off diagonal values = 0.95
## Measures of factor score adequacy             
##                                                    MR2  MR1  MR3
## Correlation of (regression) scores with factors   0.85 0.89 0.78
## Multiple R square of scores with factors          0.72 0.80 0.61
## Minimum correlation of possible factor scores     0.44 0.59 0.22

• The Cumulative proportion of variance explained by all factors dropped to 0.39.
• ‘Proportion Var’ values are equal to 0.14, 0.13 and 0.11 with all of them being bigger 0.1.
• The values of ‘Proportion Explained’ are quite similar and equal to 0.36, 0.34, 0.30.
• The root mean square of the residuals (RMSR) increased to 0.05 which is worse than it was previously since we want it to be closer to zero and at least less than 0.05.
• RMSEA index is equal to 0.111 and is now declared unacceptable.
• Tucker Lewis index of factoring reliability is equal to 0.716 which also makes it not acceptable.

The model fit according to the tests worsened. However, the plot now seems to be easier to interpret as it makes more sense.

Factor Interpretation

• MR1 - “Independance, personal space”

Such variables as “desk”, “room” and “selftab” are loaded on the first factor. One of the qualities that unites these variables and can be seen as a latent variable behind them is the independence of a child and personal space provided to him / her. It might be hypothesized that students that have higher ML1 factor scores and more personal space have higher math ahievement scores because they are not distracted by other family members and have more opportunity to concentrate on studying.

• MR2 - “Digital exposure”

“internet”, “phone” and “shared” variables loaded on the second factor that can be interpreted as the one’s level of “digital exposure”. While the presence of a personal use tablet indiates independance and is loaded on the first factor, the presence of shared tablet is loaded on the MR2 together with “phone” which is of personal use. This might seem a bit confusing, but we can hypothesize that in case of the presence of a shared tablet it’s not the “shared” quality of it that matters but the presence and access to the internet itself. Access to the internet is also loaded on this factor supporting this claim. The hypothesis can go both ways in the same way it did for the use of tablets: high “Digital exposure” factor scores might be associated with low math achievement scores because children lose their ability to concenrate and to memorize information; on the other hand, usage of the internet for educational purposes can contribute to the opposite effect.

• MR3 - “Wealth”

“dish”, “car” and “flat” variables are all, in fact, variables that were included in the dataset as indicators of wealth for Russia. Therefore, it would be natural to assume that the factor can be interpreted precisely as that. Interestingly, “game” variable is also loaded on this factor which shows that the possession of a gaming system can also be seen as an indicator of wealth. We can hypothesize that high factor “Wealth” scores are associated with higher math achievement scores because wealthy families can afford tutors and all the necessary studying materials.

Scales fit

We can now check the fit of all four scales.

## 
## Reliability analysis   
## Call: alpha(x = MR1, check.keys = TRUE)
## 
##   raw_alpha std.alpha G6(smc) average_r  S/N   ase mean   sd median_r
##       0.38      0.41    0.32      0.19 0.68 0.015 0.81 0.25     0.18
## 
##     95% confidence boundaries 
##          lower alpha upper
## Feldt     0.35  0.38  0.41
## Duhachek  0.35  0.38  0.41
## 
##  Reliability if an item is dropped:
##         raw_alpha std.alpha G6(smc) average_r  S/N alpha se var.r med.r
## desk         0.23      0.24    0.14      0.14 0.32    0.022    NA  0.14
## room         0.29      0.30    0.18      0.18 0.43    0.020    NA  0.18
## selftab      0.35      0.39    0.24      0.24 0.64    0.017    NA  0.24
## 
##  Item statistics 
##            n raw.r std.r r.cor r.drop mean   sd
## desk    4517  0.60  0.70  0.44   0.28 0.92 0.28
## room    4517  0.78  0.68  0.39   0.24 0.68 0.47
## selftab 4517  0.63  0.65  0.31   0.19 0.85 0.36
## 
## Non missing response frequency for each item
##            0    1 miss
## desk    0.08 0.92    0
## room    0.32 0.68    0
## selftab 0.15 0.85    0

## 
## Reliability analysis   
## Call: alpha(x = MR2, check.keys = TRUE)
## 
##   raw_alpha std.alpha G6(smc) average_r  S/N   ase mean   sd median_r
##       0.26      0.35    0.27      0.15 0.53 0.016 0.93 0.16     0.14
## 
##     95% confidence boundaries 
##          lower alpha upper
## Feldt     0.22  0.26  0.30
## Duhachek  0.23  0.26  0.29
## 
##  Reliability if an item is dropped:
##           raw_alpha std.alpha G6(smc) average_r  S/N alpha se var.r med.r
## internet       0.13      0.19    0.10      0.10 0.23    0.017    NA  0.10
## phone          0.20      0.24    0.14      0.14 0.32    0.019    NA  0.14
## sharedtab      0.34      0.35    0.21      0.21 0.54    0.019    NA  0.21
## 
##  Item statistics 
##              n raw.r std.r r.cor r.drop mean   sd
## internet  4517  0.55  0.68  0.40   0.20 0.97 0.18
## phone     4517  0.45  0.67  0.36   0.18 0.98 0.14
## sharedtab 4517  0.86  0.63  0.26   0.16 0.84 0.37
## 
## Non missing response frequency for each item
##              0    1 miss
## internet  0.03 0.97    0
## phone     0.02 0.98    0
## sharedtab 0.16 0.84    0

## 
## Reliability analysis   
## Call: alpha(x = MR3, check.keys = TRUE)
## 
##   raw_alpha std.alpha G6(smc) average_r  S/N   ase mean   sd median_r
##       0.43      0.43    0.37      0.16 0.76 0.014 0.41 0.28     0.15
## 
##     95% confidence boundaries 
##          lower alpha upper
## Feldt      0.4  0.43  0.45
## Duhachek   0.4  0.43  0.45
## 
##  Reliability if an item is dropped:
##      raw_alpha std.alpha G6(smc) average_r  S/N alpha se  var.r med.r
## dish      0.33      0.33    0.25      0.14 0.50    0.017 0.0044  0.13
## game      0.38      0.38    0.30      0.17 0.62    0.016 0.0014  0.16
## car       0.33      0.35    0.27      0.15 0.53    0.017 0.0054  0.14
## flat      0.38      0.39    0.30      0.17 0.63    0.016 0.0023  0.16
## 
##  Item statistics 
##         n raw.r std.r r.cor r.drop mean   sd
## dish 4517  0.59  0.63  0.41   0.27 0.21 0.40
## game 4517  0.58  0.59  0.34   0.21 0.26 0.44
## car  4517  0.63  0.62  0.39   0.26 0.68 0.47
## flat 4517  0.63  0.59  0.33   0.22 0.51 0.50
## 
## Non missing response frequency for each item
##         0    1 miss
## dish 0.79 0.21    0
## game 0.74 0.26    0
## car  0.32 0.68    0
## flat 0.49 0.51    0

Unfortunately, for all three factors standartized alpha is lower than the 0.7 treshold (0.41, 0.35, 0.43) which indicates a bad scale reliability. Moreover, when the “reliability if an item is dropped” is checked, there are no variables by dropping which higher alpha values could be reached which means that we shouldn’t get rid of any of them. Most importantly, by dropping any of the variables it would be still impossible to exceed the 0.7 treshold.