Seasonal affective disorder (SAD) is a type of depression during seasons with le
ID: 3055455 • Letter: S
Question
Seasonal affective disorder (SAD) is a type of depression during seasons with less daylight (e.g., winter months). One therapy for SAD is phototherapy, which is increased exposure to light used to improve mood. A researcher tests this therapy by exposing a sample of SAD patients to different intensities of light (low, medium, high) in a light box, either in the morning or at night (these are the times thought to be most effective for light therapy). All participants rated their mood following this therapy on a scale from 1 (poor mood) to 9 (improved mood). The hypothetical results are given in the following table.
(a) Complete the F-table and make a decision to retain or reject the null hypothesis for each hypothesis test. (Round your answers to two decimal places. Assume experimentwise alpha equal to 0.05.)
(b) Compute Tukey's HSD to analyze the significant main effect.
What is the critical value for each pairwise comparison?
(c) Summarize the results for this test using APA format.
Light Intensity Low Medium High Time ofDay Morning 5 5 7 6 6 8 4 4 6 7 7 9 5 9 5 6 8 8 Night 5 6 9 8 8 7 6 7 6 7 5 8 4 9 7 3 8 6
Explanation / Answer
There are two independent variables light intensity and time of day and dependent variable is participants rated their mood.
This is the problem of two way anova with interaction.
We can do two way anova in XLSTAT.
Steps :
ENTER data into XLSTAT sheet --> XLSTAT --> Modeling data --> ANOVA --> Y/Dependent variable --> Quantitative : select participants rated their mood variable --> X/Explanatory variable --> Qualitative --> select intensity and time of day together --> variable labels --> Options --> Interaction/ level : 2 --> Confidence level : 95 --> Outputs --> Click all the apply --> ok --> Factors and interaction --> Select all -->ok
Summary statistics (Quantitative data):
Variable Observations Obs. with missing data Obs. without missing data Minimum Maximum Mean Std. deviation
obs 36 0 36 3.000 9.000 6.500 1.595
Summary statistics (Qualitative data):
Variable Categories Counts Frequencies %
intensity 1 12 12 33.333
2 12 12 33.333
3 12 12 33.333
time of day 1 18 18 50.000
2 18 18 50.000
Correlation matrix:
intensity-1 intensity-2 intensity-3 time of day-1 time of day-2 intensity-1*time of day-1 intensity-1*time of day-2 intensity-2*time of day-1 intensity-2*time of day-2 intensity-3*time of day-1 intensity-3*time of day-2 obs
intensity-1 1 -0.500 -0.500 0.000 0.000 0.632 0.632 -0.316 -0.316 -0.316 -0.316 -0.450
intensity-2 -0.500 1 -0.500 0.000 0.000 -0.316 -0.316 0.632 0.632 -0.316 -0.316 0.150
intensity-3 -0.500 -0.500 1 0.000 0.000 -0.316 -0.316 -0.316 -0.316 0.632 0.632 0.300
time of day-1 0.000 0.000 0.000 1 -1.000 0.447 -0.447 0.447 -0.447 0.447 -0.447 -0.071
time of day-2 0.000 0.000 0.000 -1.000 1 -0.447 0.447 -0.447 0.447 -0.447 0.447 0.071
intensity-1*time of day-1 0.632 -0.316 -0.316 0.447 -0.447 1 -0.200 -0.200 -0.200 -0.200 -0.200 -0.284
intensity-1*time of day-2 0.632 -0.316 -0.316 -0.447 0.447 -0.200 1 -0.200 -0.200 -0.200 -0.200 -0.284
intensity-2*time of day-1 -0.316 0.632 -0.316 0.447 -0.447 -0.200 -0.200 1 -0.200 -0.200 -0.200 0.000
intensity-2*time of day-2 -0.316 0.632 -0.316 -0.447 0.447 -0.200 -0.200 -0.200 1 -0.200 -0.200 0.190
intensity-3*time of day-1 -0.316 -0.316 0.632 0.447 -0.447 -0.200 -0.200 -0.200 -0.200 1 -0.200 0.190
intensity-3*time of day-2 -0.316 -0.316 0.632 -0.447 0.447 -0.200 -0.200 -0.200 -0.200 -0.200 1 0.190
obs -0.450 0.150 0.300 -0.071 0.071 -0.284 -0.284 0.000 0.190 0.190 0.190 1
Regression of variable obs:
Goodness of fit statistics (obs):
Observations 36.000
Sum of weights 36.000
DF 30.000
R² 0.225
Adjusted R² 0.096
MSE 2.300
RMSE 1.517
MAPE 20.253
DW 2.161
Cp 6.000
AIC 35.421
SBC 44.922
PC 1.085
Analysis of variance (obs):
Source DF Sum of squares Mean squares F Pr > F
Model 5 20.000 4.000 1.739 0.156
Error 30 69.000 2.300
Corrected Total 35 89.000
Computed against model Y=Mean(Y)
Type I Sum of Squares analysis (obs):
Source DF Sum of squares Mean squares F Pr > F
intensity 2 18.667 9.333 4.058 0.028
time of day 1 0.444 0.444 0.193 0.663
intensity*time of day 2 0.889 0.444 0.193 0.825
Type II Sum of Squares analysis (obs):
Source DF Sum of squares Mean squares F Pr > F
intensity 2 18.667 9.333 4.058 0.028
time of day 1 0.444 0.444 0.193 0.663
intensity*time of day 2 0.889 0.444 0.193 0.825
Type III Sum of Squares analysis (obs):
Source DF Sum of squares Mean squares F Pr > F
intensity 2 18.667 9.333 4.058 0.028
time of day 1 0.444 0.444 0.193 0.663
intensity*time of day 2 0.889 0.444 0.193 0.825
Model parameters (obs):
Source Value Standard error t Pr > |t| Lower bound (95%) Upper bound (95%)
Intercept 7.167 0.619 11.575 < 0.0001 5.902 8.431
intensity-1 -1.667 0.876 -1.903 0.067 -3.455 0.122
intensity-2 0.000 0.876 0.000 1.000 -1.788 1.788
intensity-3 0.000 0.000
time of day-1 0.000 0.876 0.000 1.000 -1.788 1.788
time of day-2 0.000 0.000
intensity-1*time of day-1 0.000 1.238 0.000 1.000 -2.529 2.529
intensity-1*time of day-2 0.000 0.000
intensity-2*time of day-1 -0.667 1.238 -0.538 0.594 -3.196 1.862
intensity-2*time of day-2 0.000 0.000
intensity-3*time of day-1 0.000 0.000
intensity-3*time of day-2 0.000 0.000
Equation of the model (obs):
obs = 7.16666666666667-1.66666666666666*intensity-1-0.666666666666667*intensity-2*time of day-1
Standardized coefficients (obs):
Source Value Standard error t Pr > |t| Lower bound (95%) Upper bound (95%)
intensity-1 -0.500 0.263 -1.903 0.067 -1.036 0.036
intensity-2 0.000 0.263 0.000 1.000 -0.536 0.536
intensity-3 0.000 0.000
time of day-1 0.000 0.278 0.000 1.000 -0.569 0.569
time of day-2 0.000 0.000
intensity-1*time of day-1 0.000 0.294 0.000 1.000 -0.599 0.599
intensity-1*time of day-2 0.000 0.000
intensity-2*time of day-1 -0.158 0.294 -0.538 0.594 -0.757 0.441
intensity-2*time of day-2 0.000 0.000
intensity-3*time of day-1 0.000 0.000
intensity-3*time of day-2 0.000 0.000
Predictions and residuals (obs):
Observation Weight obs Pred(obs) Residual Std. residual Std. dev. on pred. (Mean) Lower bound 95% (Mean) Upper bound 95% (Mean) Std. dev. on pred. (Observation) Lower bound 95% (Observation) Upper bound 95% (Observation)
Obs1 1 5.000 5.500 -0.500 -0.330 0.619 4.236 6.764 1.638 2.155 8.845
Obs2 1 6.000 5.500 0.500 0.330 0.619 4.236 6.764 1.638 2.155 8.845
Obs3 1 4.000 5.500 -1.500 -0.989 0.619 4.236 6.764 1.638 2.155 8.845
Obs4 1 7.000 5.500 1.500 0.989 0.619 4.236 6.764 1.638 2.155 8.845
Obs5 1 5.000 5.500 -0.500 -0.330 0.619 4.236 6.764 1.638 2.155 8.845
Obs6 1 6.000 5.500 0.500 0.330 0.619 4.236 6.764 1.638 2.155 8.845
Obs7 1 5.000 5.500 -0.500 -0.330 0.619 4.236 6.764 1.638 2.155 8.845
Obs8 1 8.000 5.500 2.500 1.648 0.619 4.236 6.764 1.638 2.155 8.845
Obs9 1 6.000 5.500 0.500 0.330 0.619 4.236 6.764 1.638 2.155 8.845
Obs10 1 7.000 5.500 1.500 0.989 0.619 4.236 6.764 1.638 2.155 8.845
Obs11 1 4.000 5.500 -1.500 -0.989 0.619 4.236 6.764 1.638 2.155 8.845
Obs12 1 3.000 5.500 -2.500 -1.648 0.619 4.236 6.764 1.638 2.155 8.845
Obs13 1 5.000 6.500 -1.500 -0.989 0.619 5.236 7.764 1.638 3.155 9.845
Obs14 1 6.000 6.500 -0.500 -0.330 0.619 5.236 7.764 1.638 3.155 9.845
Obs15 1 4.000 6.500 -2.500 -1.648 0.619 5.236 7.764 1.638 3.155 9.845
Obs16 1 7.000 6.500 0.500 0.330 0.619 5.236 7.764 1.638 3.155 9.845
Obs17 1 9.000 6.500 2.500 1.648 0.619 5.236 7.764 1.638 3.155 9.845
Obs18 1 8.000 6.500 1.500 0.989 0.619 5.236 7.764 1.638 3.155 9.845
Obs19 1 6.000 7.167 -1.167 -0.769 0.619 5.902 8.431 1.638 3.821 10.512
Obs20 1 8.000 7.167 0.833 0.549 0.619 5.902 8.431 1.638 3.821 10.512
Obs21 1 7.000 7.167 -0.167 -0.110 0.619 5.902 8.431 1.638 3.821 10.512
Obs22 1 5.000 7.167 -2.167 -1.429 0.619 5.902 8.431 1.638 3.821 10.512
Obs23 1 9.000 7.167 1.833 1.209 0.619 5.902 8.431 1.638 3.821 10.512
Obs24 1 8.000 7.167 0.833 0.549 0.619 5.902 8.431 1.638 3.821 10.512
Obs25 1 7.000 7.167 -0.167 -0.110 0.619 5.902 8.431 1.638 3.821 10.512
Obs26 1 8.000 7.167 0.833 0.549 0.619 5.902 8.431 1.638 3.821 10.512
Obs27 1 6.000 7.167 -1.167 -0.769 0.619 5.902 8.431 1.638 3.821 10.512
Obs28 1 9.000 7.167 1.833 1.209 0.619 5.902 8.431 1.638 3.821 10.512
Obs29 1 5.000 7.167 -2.167 -1.429 0.619 5.902 8.431 1.638 3.821 10.512
Obs30 1 8.000 7.167 0.833 0.549 0.619 5.902 8.431 1.638 3.821 10.512
Obs31 1 9.000 7.167 1.833 1.209 0.619 5.902 8.431 1.638 3.821 10.512
Obs32 1 7.000 7.167 -0.167 -0.110 0.619 5.902 8.431 1.638 3.821 10.512
Obs33 1 6.000 7.167 -1.167 -0.769 0.619 5.902 8.431 1.638 3.821 10.512
Obs34 1 8.000 7.167 0.833 0.549 0.619 5.902 8.431 1.638 3.821 10.512
Obs35 1 7.000 7.167 -0.167 -0.110 0.619 5.902 8.431 1.638 3.821 10.512
Obs36 1 6.000 7.167 -1.167 -0.769 0.619 5.902 8.431 1.638 3.821 10.512
Interpretation (obs):
Given the R2, 22% of the variability of the dependent variable obs is explained by the 3 explanatory variables.
Given the p-value of the F statistic computed in the ANOVA table, and given the significance level of 5%, the information brought by the explanatory variables is not significantly better than what a basic mean would bring. The fact that variables do not bring significant information to the model may be interpreted in different ways: Either the variables do not contribute to the model, or some covariates that would help explaining the variability are missing, or the model is wrong, or the data contain errors.
Based on the Type III sum of squares, the following variables bring significant information to explain the variability of the dependent variable obs: intensity.
Based on the Type III sum of squares, the following variables do not bring significant information to explain the variability the dependent variable obs: time of day,intensity*time of day. You might want to remove them from the model.
Among the explanatory variables, based on the Type III sum of squares, variable intensity is the most influential.
LS Means for factor intensity:
Category LS mean Standard error Lower bound (95%) Upper bound (95%)
1 5.500 0.438 4.606 6.394
2 6.833 0.438 5.939 7.727
3 7.167 0.438 6.273 8.061
intensity*time of day:
Category LS mean Standard error Lower bound (95%) Upper bound (95%)
intensity-1*time of day-1 5.500 0.619 4.236 6.764
intensity-1*time of day-2 5.500 0.619 4.236 6.764
intensity-2*time of day-1 6.500 0.619 5.236 7.764
intensity-2*time of day-2 7.167 0.619 5.902 8.431
intensity-3*time of day-1 7.167 0.619 5.902 8.431
intensity-3*time of day-2 7.167 0.619 5.902 8.431
intensity ime of day 1 2
1 5.500 5.500
2 6.500 7.167
3 7.167 7.167
LS Means for factor time of day:
Category LS mean Standard error Lower bound (95%) Upper bound (95%)
1 6.389 0.357 5.659 7.119
2 6.611 0.357 5.881 7.341
time of day*intensity:
Category LS mean Standard error Lower bound (95%) Upper bound (95%)
intensity-1*time of day-1 5.500 0.619 4.236 6.764
intensity-1*time of day-2 5.500 0.619 4.236 6.764
intensity-2*time of day-1 6.500 0.619 5.236 7.764
intensity-2*time of day-2 7.167 0.619 5.902 8.431
intensity-3*time of day-1 7.167 0.619 5.902 8.431
intensity-3*time of day-2 7.167 0.619 5.902 8.431
time of dayintensity 1 2 3
1 5.500 6.500 7.167
2 5.500 7.167 7.167
obs intensity time of day 5 1 1 6 1 1 4 1 1 7 1 1 5 1 1 6 1 1 5 1 2 8 1 2 6 1 2 7 1 2 4 1 2 3 1 2 5 2 1 6 2 1 4 2 1 7 2 1 9 2 1 8 2 1 6 2 2 8 2 2 7 2 2 5 2 2 9 2 2 8 2 2 7 3 1 8 3 1 6 3 1 9 3 1 5 3 1 8 3 1 9 3 2 7 3 2 6 3 2 8 3 2 7 3 2 6 3 2Related Questions
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