The average time for differen plans by Grade

##                              plan_name Grade mean   n
## 1                          Custom Plan    PK  1.0   1
## 2                          Custom Plan    PS  2.0   2
## 3               T1 Addition Strategies    PK  3.0  54
## 4               T1 Addition Strategies    PS  2.5  22
## 5            T1 Classification/Sorting    PK  2.0   4
## 6                 T1 Comparative Value    PK  2.0  20
## 7                 T1 Comparative Value    PS  1.8  19
## 8                    T1 Compound Words    PK  2.3  34
## 9                    T1 Compound Words    PS  2.6  77
## 10                    T1 Conversations    PK  2.1  39
## 11                    T1 Conversations    PS  2.0  33
## 12                T1 ELL Units 2-4 DIP    PK  3.1   5
## 13                T1 ELL Units 2-4 DIP    PS  2.7   9
## 14               T1 ELL Units 9-10 DIP    PK  1.2   2
## 15               T1 ELL Units 9-10 DIP    PS  1.4   6
## 16         T1 Expression: Ask & Answer    PK  2.0  12
## 17         T1 Expression: Ask & Answer    PS  2.0  28
## 18 T1 Expression: Descriptive Language    PK  2.3  76
## 19 T1 Expression: Descriptive Language    PS  2.2  90
## 20                     T1 Final Sounds    PK  1.8  13
## 21                   T1 Initial Sounds    PK  2.4  18
## 22                   T1 Initial Sounds    PS  1.9  11
## 23             T1 Letter Sounds and ID    PK  1.0   9
## 24             T1 Letter Sounds and ID    PS  1.0   1
## 25          T1 Narrative Comprehension    PK  1.0   4
## 26                   T1 Number Symbols    PK  3.0  76
## 27                   T1 Number Symbols    PS  2.5  67
## 28                       T1 One-to-One    PK  2.7  39
## 29                       T1 One-to-One    PS  3.0  70
## 30               T1 Oral Comprehension    PK  2.5  33
## 31               T1 Oral Comprehension    PS  2.6  73
## 32                         T1 Ordering    PK  3.0   1
## 33                         T1 Patterns    PK  2.5   2
## 34                         T1 Phonemes    PK  3.0  89
## 35                         T1 Phonemes    PS  2.6  69
## 36                   T1 Quantification    PK  2.4  79
## 37                   T1 Quantification    PS  2.4  70
## 38                T1 Rational Counting    PK  3.0 132
## 39                T1 Rational Counting    PS  3.4 128
## 40                            T1 Rhyme    PK  3.4 115
## 41                            T1 Rhyme    PS  3.3 184
## 42                    T1 Rote Counting    PK  2.7  51
## 43                    T1 Rote Counting    PS  2.2  66
## 44                 T1 SED PK Units 1-5    PK  3.1  41
## 45                 T1 SED PS Units 1-5    PK  5.2   2
## 46                 T1 SED PS Units 1-5    PS  3.3  58
## 47                T1 SED PS Units 6-10    PK  4.0   2
## 48                T1 SED PS Units 6-10    PS  3.5  21
## 49             T1 Shape Identification    PK  2.5   6
## 50             T1 Shape Identification    PS  2.5   1
## 51                T1 Spatial Awareness    PK  1.5   1
## 52           T1 Subtraction Strategies    PK  2.9  22
## 53           T1 Subtraction Strategies    PS  2.6   9
## 54                        T1 Syllables    PK  2.7  47
## 55                        T1 Syllables    PS  2.7  60
## 56                       T1 Vocabulary    PK  1.8   4
## 57                       T1 Vocabulary    PS  1.1   5
## 58                   T1 Word Awareness    PK  3.4  66
## 59                   T1 Word Awareness    PS  3.3  76

note: I first define diff as the value of Round_Last_lesson_Taught minus its corresponding average time. if the absolute value of diff is less than 0.3, then I define rate as equal, otherwise, if diff>0, I define rate as more and if diff<0, I define rate as less.

Build linear regressions to study the relation between the lesson pace and PassRate

model1= lm(PassRate~rate+Grade, data=pace1)
summary(model1)
## 
## Call:
## lm(formula = PassRate ~ rate + Grade, data = pace1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.8339 -0.1515  0.1661  0.2363  0.2822 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.83389    0.01937  43.044  < 2e-16 ***
## rateless    -0.04596    0.02360  -1.947   0.0517 .  
## ratemore    -0.01571    0.02088  -0.753   0.4518    
## GradePS     -0.07015    0.01759  -3.988 7.07e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.297 on 1161 degrees of freedom
## Multiple R-squared:  0.01525,    Adjusted R-squared:  0.01271 
## F-statistic: 5.994 on 3 and 1161 DF,  p-value: 0.0004717
plot(model1,which=3)

The above result shows that both rate and Grade have signficant impact on the PassRate, then I check the linear model assumpution by ploting the fitted value towards residul. In the plot,it doesn’t satisfy the linear model assumption of constant residual variance because the red line in the plot is not constant

some viuslization

According to the above correlation matrix, none of the numerical variables are highly linear correlated with PassRate

Build spline model (non-linear)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## PassRate ~ s(Last_Lesson_Taught)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.777867   0.008673   89.69   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                         edf Ref.df     F  p-value    
## s(Last_Lesson_Taught) 2.582  3.239 6.952 7.92e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.0188   Deviance explained =  2.1%
## GCV = 0.087908  Scale est. = 0.087637  n = 1165

According to the result, the spline item of Last_Lesson_Taught significant affects on PassRate. The plot shows what kind of transformation applies on Last_Lesson_Taught.