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Linear Regression, Essay Example
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Independent T-Test Samples
Independent Samples Test | ||||||||
Levene’s Test for Equality of Variances | t-test for Equality of Means | |||||||
F | Sig. | t | df | Sig. (2-tailed) | Mean Difference | Std. Error Difference | ||
CTBS Language NCE score | Equal variances assumed | .597 | .440 | 3.016 | 314 | .003 | 6.01259 | 1.99351 |
Equal variances not assumed | 3.012 | 310.703 | .003 | 6.01259 | 1.99616 |
One-Way Anova
ANOVA | |||||
CTBS Language NCE score | |||||
Sum of Squares | df | Mean Square | F | Sig. | |
Between Groups | 2854.121 | 1 | 2854.121 | 9.097 | .003 |
Within Groups | 98517.691 | 314 | 313.751 | ||
Total | 101371.812 | 315 |
General Linear Regression
Model Summary | ||||
Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |
1 | .168a | .028 | .025 | 17.71301 |
a. Predictors: (Constant), gender |
Coefficientsa | ||||||
Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
B | Std. Error | Beta | ||||
1 | (Constant) | 52.994 | 1.392 | 38.080 | .000 | |
gender | -6.013 | 1.994 | -.168 | -3.016 | .003 | |
a. Dependent Variable: CTBS Language NCE score |
The t-test is simply comparing the language score means between males and females based on the correlational coefficient. In this example, the female mean score (52.990) is higher than the male mean score (46.98), when the size of the sample is taken into account, a t-score of 3.01 is calculated (p-value>.001).
One-way ANOVA is also a correlational analysis, based on the correlational coefficient between variables. The ANOVA analysis comes up with the exact same p-value as the t-test (.003); however, it uses an f-test rather than t-test.
General linear regression gives substantially more information than a t-test or ANOVA. The parameter “gender” is significant ( a face one would know from the previous two tests); however, the regression allows to know the correlation between the two (negative) and the approximate effect of gender on tests.
Univariate Analysis
Tests of Between-Subjects Effects | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dependent Variable:CTBS Language NCE score | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Source | Type III Sum of Squares | df | Mean Square | F | Sig. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Corrected Model | 11678.496a | 9 | 1297.611 | 4.427 | .000 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Intercept | 122302.969 | 1 | 122302.969 | 417.252 | .000 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
gender | 42.755 | 1 | 42.755 | .146 | .703 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
engprof | 8410.914 | 4 | 2102.728 | 7.174 | .000 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
gender * engprof | 1263.217 | 4 | 315.804 | 1.077 | .368 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Error | 89693.316 | 306 | 293.115 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Total | 893388.982 | 316 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Corrected Total | 101371.812 | 315 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Multiple Regression
The main difference between the two models is that the unvariate analysis gives on whether the factor (gender or bilingual) was a significant predictor of the language score (via the F-score). The linear regression model actually gives parameter estimates for the main factors (gender, proficiency. Due to the presence of categorical variables, this interpretation is not intuitive. Gender (-.194) is actually the parameter for being male (“male=1”) meaning that being male lowers one test scores. Similarly, there is a negative interpretation of proficiency in native language variable. Finally, the interaction variable is significant. |
The average mark in algebra (dependent variable) is skewed, largely because of the decision to group the variables by grades rather than keeping the continuous scores- a large number of students that failed (112) and earned a d, but the decision to model the variable in this way essentially puts high fails and low “d”s in the same category. The CTBS Math NCE score looks to be roughly normally distributed- based on the decision to keep it as a continuous score. The “school” variable is even distributed between school alpha and school beta. Finally, number of days absent is positively skewed: this is a function of numerous students not missing any days of school (0), and the distribution thinning as the number of absent days increases. Although there are some problems with this data, the regression is generally robust.
Coefficientsa | ||||||
Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
B | Std. Error | Beta | ||||
1 | (Constant) | .728 | .236 | 3.079 | .002 | |
number of days absent | -.047 | .009 | -.271 | -5.376 | .000 | |
CTBS Math percentile | .022 | .003 | .439 | 8.592 | .000 | |
mark in spring math | .137 | .051 | .135 | 2.670 | .008 | |
school | -.310 | .134 | -.121 | -2.307 | .022 | |
a. Dependent Variable: mark in algebra |
Looking at the results, all of the parameters are significant, and most of the results make intuitive sense: an increase in the number of days absent leads to a lower algebra score; grades in the CTBS and mark in spring math are positively correlated with the score in algebra; finally, depending on the school chosen, there is a negative correlation. Looking at the effect size, the mark in spring math is a strong positive predictor; and school a strong negative predictor.
The difference between unstandardized coefficients and standardized coefficients is the use of scale: standardized coefficients are corrected for scale. The main difference between the two are the strong negative predictor (number of days absent) and the strong positive predictor (CTBS Math Percentile). Finally, school is less important once scaled to the other variables
I chose to add on the model above to understand how variables that look at English grades and bilingual status might also predict math grades. The intuition behind this inclusion is that individuals that don’t understand English may not understand what the math instructor is saying; or may also not ask needed questions due to a perceived obstacle in communication with the teacher.
Model Summary | ||||
Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |
1 | .660a | .436 | .425 | .973 |
a. Predictors: (Constant), number of days absent, profiiency in native language, CTBS Math NCE score, mark in English 95, bilingual status, CTBS Language NCE score |
Model= Algebra Scores (DV)= Mark in English (IV)+ CTBS Math Score (IV)+ CTBS Language NCE Score (IV) + Bilingual status (IV)+ Proficiency in native language (IV)+ Number of days absent
Coefficientsa | ||||||
Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
B | Std. Error | Beta | ||||
1 | (Constant) | -.504 | .233 | -2.165 | .031 | |
mark in English 95 | .458 | .052 | .448 | 8.793 | .000 | |
CTBS Math NCE score | .027 | .004 | .377 | 6.367 | .000 | |
CTBS Language NCE score | -.008 | .004 | -.108 | -1.738 | .083 | |
bilingual status | -.024 | .060 | -.021 | -.407 | .684 | |
profiiency in native language | .126 | .059 | .108 | 2.138 | .033 | |
number of days absent | -.021 | .008 | -.124 | -2.623 | .009 | |
a. Dependent Variable: mark in algebra |
Overall, this model has a significantly higher adjusted r-squared than the previous model- this is important because the regular r-squared will increase regardless of whether the variable is significant or not. My hypothesis that language plays a role in math scores had a mixed result: “Mark in English 95” was a significant positive predictor of the algebra score; this could be because it is capturing that good students do well, or it could be that those with better English ability can understand math better. “Proficiency in native language” was also a significant positive predictor leading one to believe that language ability plays some type of role in understanding math. The other two variables, “CTBS language score” and “Bilingual Status” were not significant in predicting the algebra score.
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