The dataset “math.sav” contains math scores for kids assessed at 6 different grade levels and an indicator of whether it was a boy (1) or a girl (2) [variable “gender”]. Carry out analyses to evaluate developmental and gender effects. Make sure to state research questions and hypotheses, specify appropriate statistics, test model assumptions, present and interpret relevant statistical and graphical results in the APA style and conclude with a general statement that could be understood by an educated reader who is not versed in statistics.

math_df <- haven::read_sav(file = "math.sav")

Data Exploration

Summary Stats

# math_df %>% str
math_df$gender %<>% factor(labels = c(`1` = "M", `2` = "F"))
tags$p("Summary statistics by Male and Female genders respectively are below:")

Summary statistics by Male and Female genders respectively are below:

psych::describeBy(math_df %>% select(-cid, -gender), list(math_df$gender)) %>% purrr::walk(.f = function(.) stargazer::stargazer(., 
    summary = F, type = "html", rownames = T, colnames = T, header = F, digits = 3))
vars n mean sd median trimmed mad min max range skew kurtosis se
math_1 1 101 26.145 8.223 25.360 25.660 7.087 10.670 51.370 40.700 0.579 0.078 0.818
math_2 2 101 38.575 13.334 36.240 37.227 10.971 15.960 78.080 62.120 0.894 0.490 1.327
math_3 3 101 45.553 14.019 44.290 44.683 12.202 18.390 86.120 67.730 0.632 0.477 1.395
math_4 4 101 62.100 18.558 60.320 60.929 17.584 31.280 109.300 78.020 0.527 -0.377 1.847
math_8 5 101 97.389 20.311 101.480 98.597 21.379 48.790 132.040 83.250 -0.478 -0.473 2.021
math_12 6 101 119.242 19.468 123.500 121.381 16.575 60.670 148.850 88.180 -0.920 0.380 1.937
vars n mean sd median trimmed mad min max range skew kurtosis se
math_1 1 99 23.203 9.149 21.240 22.237 7.161 9.480 63.220 53.740 1.664 4.606 0.920
math_2 2 99 32.391 11.128 30.240 31.344 9.251 13.640 65.060 51.420 0.889 0.560 1.118
math_3 3 99 39.504 12.928 37.710 38.306 9.993 14.470 75.490 61.020 0.865 0.606 1.299
math_4 4 99 56.812 15.860 55.300 56.167 14.233 19.950 96.840 76.890 0.430 -0.188 1.594
math_8 5 99 89.142 21.246 88.280 89.076 19.407 35.770 138.670 102.900 0.013 -0.263 2.135
math_12 6 99 111.522 20.926 113.320 112.917 20.371 62.620 144.800 82.180 -0.524 -0.400 2.103 
tags$p("The numbers of individuals in groups are equal. The math scores at time point 1 may not be normally distributed.")

The numbers of individuals in groups are equal. The math scores at time point 1 may not be normally distributed.

Graphical Exploration

g <- lapply(math_df %>% select(starts_with("math")) %>% names, function(nm) {
    math_df %>% ggplot(data = ., mapping = aes_string(x = nm)) + geom_histogram(binwidth = 2, 
        aes(y = ..density..)) + geom_density(col = 2) + facet_grid(~gender)
})
gridExtra::grid.arrange(grobs = g, nrow = 3)

tags$p("Time points 1 and 2, as expected from the summary statistics, exhibit right skew and show a couple of outliers. The graphs over time show a progression from right skew to semi-normality to left skew suggesting the gradual improvement over time in math scores. We can evaluate this observation by averaging the data and looking at trend lines.")

Time points 1 and 2, as expected from the summary statistics, exhibit right skew and show a couple of outliers. The graphs over time show a progression from right skew to semi-normality to left skew suggesting the gradual improvement over time in math scores. We can evaluate this observation by averaging the data and looking at trend lines.

math_df %>% select(-cid) %>% group_by(gender) %>% summarize_all(.funs = mean) %>% 
    as.data.frame() %>% reshape(varying = names(math_df)[-c(1, 2)], v.names = "avg", 
    timevar = "yr", times = names(math_df)[-c(1, 2)], direction = "long") %>% mutate(s = str_extract(yr, 
    "\\d{1,2}") %>% as.numeric) %>% arrange(s) %T>% {
    .$yr <- factor(.$yr, levels = unique(.$yr[order(.$s)]))
} %T>% {
    .$gender %<>% factor(labels = c(`1` = "M", `2` = "F"))
} %>% ggplot(data = ., mapping = aes(y = avg, x = yr, color = gender, group = gender)) + 
    geom_line() + geom_smooth(method = "lm", alpha = 0.2) + labs(title = "Average Scores across Time", 
    subtitle = "", caption = "", x = "Time Points", y = "Average Score") + theme(plot.title = element_text(hjust = 0.5), 
    plot.subtitle = element_text(hjust = 0.5))

tags$p("As anticipated, there is a significant positive correlation between time and the math score. Gender does not appear to be significant, but the ANOVA is more suited to make this determination..")

As anticipated, there is a significant positive correlation between time and the math score. Gender does not appear to be significant, but the ANOVA is more suited to make this determination..

Hypotheses

\[\begin{aligned} \text{There is no change across subjects by gender: }& \hat{y}_{M} = \hat{y}_{F} \\ \text{There is no change between subjects across time: }& \hat{y}_{t_1} = \hat{y}_{t_2} ...=... \hat{y}_{t_x} \\ \text{There is no change due to interaction effects of gender and time: }& \hat{y}^{M}_{t_1} = \hat{y}^{F}_{t_1} ... = ... \hat{y}^{M}_{t_x} = \hat{y}^{F}_{t_x} \end{aligned}\]

Assumption Testing

Normality

The summary statistics make it evident that violations of normality are present in at least the first time observation and likely the last time observation, but given the nature of measuring math scores this is to be expected.

Sphericity ?

Repeated-Measures ANOVA

math_df.long <- math_df %>% tidyr::gather(key = "time", value = list(math = math_1:math_12), 
    -gender, -cid, factor_key = T) %>% rename(math = 4)

ANOVA Simple Version

anova(lm(math ~ time + gender + (time * gender)^2, data = math_df.long)) %T>% assign("math_long.aov", 
    ., envir = .GlobalEnv) %>% stargazer::stargazer(summary = F, type = "html", rownames = T, 
    colnames = T, header = F, digits = 3)
Df Sum Sq Mean Sq F value Pr(> F)
time 5 1,262,337.000 252,467.400 979.192 0
gender 1 11,059.060 11,059.060 42.892 0
time:gender 5 893.274 178.655 0.693 0.629
Residuals 1,188 306,305.000 257.833 
# https://stackoverflow.com/questions/35104052/how-to-remake-aov-to-car-package-anova-to-get-mauchlys-test-for-sphericity
# car::Anova(lm(math~time+gender+(time*gender)^2,idata =
# data.frame(gender=math_df.long$gender,math=math_df.long$math),idesign=~time*gender,data=math_df.long),type=3)
# aov(math ~ gender * time + Error(gender/time),data=math_df.long) %>%
# summary.aov

The results indicate significance both within subjects across grade levels F (5) =979.19, p<.001 and between subjects across gender F (1) =42.89, p<.001, but no interaction between the grade levels.

However, we might be interested on math outcomes at the conclusion of primary school. We can look all combinations of possible interactions between gender and grade levels on the math scores in grade 12.

ANOVA All possible Interactions

paste("math_12~", "+gender+", paste0(math_df %>% select(starts_with("math")) %>% 
    names %>% .[-length(.)], collapse = "+"), "+(", paste0(math_df %>% select(starts_with("math")) %>% 
    names %>% .[-length(.)], collapse = "+"), "+gender)^7", sep = "") %>% as.formula %>% 
    lm(data = math_df) %>% anova() %T>% assign("math_aov", ., envir = .GlobalEnv) %>% 
    stargazer::stargazer(summary = F, type = "html", rownames = T, colnames = T, 
        header = F, digits = 3)
Df Sum Sq Mean Sq F value Pr(> F)
gender 1 2,979.206 2,979.206 35.643 0.00000
math_1 1 30,059.270 30,059.270 359.629 0
math_2 1 5,577.759 5,577.759 66.732 0
math_3 1 6,842.475 6,842.475 81.863 0
math_4 1 5,959.024 5,959.024 71.294 0
math_8 1 16,577.280 16,577.280 198.330 0
math_1:math_2 1 873.738 873.738 10.453 0.002
math_1:math_3 1 565.361 565.361 6.764 0.010
math_1:math_4 1 0.020 0.020 0.0002 0.988
math_1:math_8 1 66.884 66.884 0.800 0.373
gender:math_1 1 107.013 107.013 1.280 0.260
math_2:math_3 1 116.006 116.006 1.388 0.241
math_2:math_4 1 270.144 270.144 3.232 0.074
math_2:math_8 1 34.018 34.018 0.407 0.525
gender:math_2 1 10.564 10.564 0.126 0.723
math_3:math_4 1 97.983 97.983 1.172 0.281
math_3:math_8 1 135.374 135.374 1.620 0.205
gender:math_3 1 5.105 5.105 0.061 0.805
math_4:math_8 1 15.410 15.410 0.184 0.668
gender:math_4 1 80.079 80.079 0.958 0.329
gender:math_8 1 43.907 43.907 0.525 0.470
math_1:math_2:math_3 1 16.431 16.431 0.197 0.658
math_1:math_2:math_4 1 2.169 2.169 0.026 0.872
math_1:math_2:math_8 1 55.019 55.019 0.658 0.419
gender:math_1:math_2 1 0.973 0.973 0.012 0.914
math_1:math_3:math_4 1 11.320 11.320 0.135 0.713
math_1:math_3:math_8 1 34.100 34.100 0.408 0.524
gender:math_1:math_3 1 3.778 3.778 0.045 0.832
math_1:math_4:math_8 1 23.061 23.061 0.276 0.600
gender:math_1:math_4 1 27.191 27.191 0.325 0.569
gender:math_1:math_8 1 66.106 66.106 0.791 0.375
math_2:math_3:math_4 1 2.802 2.802 0.034 0.855
math_2:math_3:math_8 1 178.046 178.046 2.130 0.147
gender:math_2:math_3 1 6.316 6.316 0.076 0.784
math_2:math_4:math_8 1 60.490 60.490 0.724 0.396
gender:math_2:math_4 1 6.604 6.604 0.079 0.779
gender:math_2:math_8 1 27.267 27.267 0.326 0.569
math_3:math_4:math_8 1 23.918 23.918 0.286 0.594
gender:math_3:math_4 1 1.004 1.004 0.012 0.913
gender:math_3:math_8 1 395.426 395.426 4.731 0.031
gender:math_4:math_8 1 32.679 32.679 0.391 0.533
math_1:math_2:math_3:math_4 1 21.389 21.389 0.256 0.614
math_1:math_2:math_3:math_8 1 62.383 62.383 0.746 0.389
gender:math_1:math_2:math_3 1 17.469 17.469 0.209 0.648
math_1:math_2:math_4:math_8 1 0.252 0.252 0.003 0.956
gender:math_1:math_2:math_4 1 8.015 8.015 0.096 0.757
gender:math_1:math_2:math_8 1 0.626 0.626 0.007 0.931
math_1:math_3:math_4:math_8 1 13.336 13.336 0.160 0.690
gender:math_1:math_3:math_4 1 22.650 22.650 0.271 0.604
gender:math_1:math_3:math_8 1 19.265 19.265 0.230 0.632
gender:math_1:math_4:math_8 1 65.717 65.717 0.786 0.377
math_2:math_3:math_4:math_8 1 69.702 69.702 0.834 0.363
gender:math_2:math_3:math_4 1 4.052 4.052 0.048 0.826
gender:math_2:math_3:math_8 1 7.349 7.349 0.088 0.767
gender:math_2:math_4:math_8 1 127.012 127.012 1.520 0.220
gender:math_3:math_4:math_8 1 32.594 32.594 0.390 0.533
math_1:math_2:math_3:math_4:math_8 1 61.319 61.319 0.734 0.393
gender:math_1:math_2:math_3:math_4 1 276.169 276.169 3.304 0.071
gender:math_1:math_2:math_3:math_8 1 128.143 128.143 1.533 0.218
gender:math_1:math_2:math_4:math_8 1 53.071 53.071 0.635 0.427
gender:math_1:math_3:math_4:math_8 1 1.780 1.780 0.021 0.884
gender:math_2:math_3:math_4:math_8 1 43.228 43.228 0.517 0.473
gender:math_1:math_2:math_3:math_4:math_8 1 0.068 0.068 0.001 0.977
Residuals 136 11,367.440 83.584 
sl <- c(0.001, 0.01, 0.05, 0.1)
math_aov.sig <- math_aov[math_aov$`Pr(>F)` < 0.1, ] %>% na.omit
for (i in seq_along(sl)) {
    cat("Variables with a significant influence on grade 12 scores at the ", sl[i], 
        " alpha level include: \n", sep = "\n")
    
    if (i == 1) {
        l <- math_aov$`Pr(>F)` < sl[i]
    } else {
        l <- {
            math_aov$`Pr(>F)` < sl[i] & math_aov$`Pr(>F)` >= sl[i - 1]
        }
    }
    math_aov.sig <- math_aov[l, ] %>% na.omit
    for (rn in 1:nrow(math_aov.sig)) {
        paste(row.names(math_aov.sig)[rn], math_aov.sig[rn, ] %>% apa_t(type = "html"), 
            "\n") %>% cat(sep = "\n")
    }
}

Variables with a significant influence on grade 12 scores at the 0.001 alpha level include:

gender F (1) =35.64, p<.001

math_1 F (1) =359.63, p<.001

math_2 F (1) =66.73, p<.001

math_3 F (1) =81.86, p<.001

math_4 F (1) =71.29, p<.001

math_8 F (1) =198.33, p<.001

Variables with a significant influence on grade 12 scores at the 0.01 alpha level include:

math_1:math_2 F (1) =10.45, p<.01

Variables with a significant influence on grade 12 scores at the 0.05 alpha level include:

math_1:math_3 F (1) =6.76, p<.05

gender:math_3:math_8 F (1) =4.73, p<.05

Variables with a significant influence on grade 12 scores at the 0.1 alpha level include:

math_2:math_4 F (1) =3.23, p<.1

gender:math_1:math_2:math_3:math_4 F (1) =3.3, p<.1

cat("Gender is again a significant influencing variable on grade 12 math scores, as are scores in each grade level prior to grade 12. There are also significant interaction effects, the most significant of which are interactions between early grade level math scores, and the grades across gender for grades 3 & 8. These finding suggest that the early exposure to math in grades 1-3 has a more significant impact on how students perform as they graduate high school. It also suggests that males and females may learn different in grades 3 and grade 8 due to some developmental effects. Further research would be necessary to determine the nature of this interaction.")

Gender is again a significant influencing variable on grade 12 math scores, as are scores in each grade level prior to grade 12. There are also significant interaction effects, the most significant of which are interactions between early grade level math scores, and the grades across gender for grades 3 & 8. These finding suggest that the early exposure to math in grades 1-3 has a more significant impact on how students perform as they graduate high school. It also suggests that males and females may learn different in grades 3 and grade 8 due to some developmental effects. Further research would be necessary to determine the nature of this interaction.