Present study is aimed to test whether daily exercise enhance the memory in people with moderate dementia. For this study patients have been chosen randomly and out of total sample 20 has undergone half an hour of supervised treadmill waling at the university on daily basis for a month. 20 people out of sample were asked to visit the university and have conversation with the researcher for half an hour each day for a month. All the relevant data related to the memory has been stored by the researcher before the treatment period, during the test and at the end of the treatment. Each participant had scope to achieve highest marks of 100 and researcher scored them as per the memory test after the complete test. Considering the achieved marks statistical analysis has been done and outcome has been presented below.
For the present study, two way mixed Anova method has been utilised because there were two independent variable. As per the stored data, time points and exercise levels were the two independent variables. Besides this time has been considered as a variable for the study and it is repeated measure that means participants in all three conditions of the Anova test were dependent. Other four conditions of the Anova were independent measures where different participants were contested under different conditions.
The analysis has been done meeting all the assumptions for the two way mixed anova. As per the first assumption of two way mixed Anova method, dependent variable should be measured at a continuous level. As per the present case, dependent variable (Memory), has been set to continuous level so as to meet the first assumption. Second assumption of the two way mixed Anova method depicts that within subject factors, there should be at least two categorical matched groups that represent same subjects within both groups. As per the present case, there were two categorical variable that makes the test justified with second assumption as well.
WithinSubjects Factors 

Measure: MEASURE_1 

health 
Dependent Variable 
1 
Baseline 
2 
Two_weeks 
3 
Four_weeks 
BetweenSubjects Factors 


N 

Group 
Exercise 
19 
No_exercise 
20 
Thirdly two way mixed Anova method need to have at least two independent groups between subjects factor independent variable. As per the group distribution it can be seen that there has been no significant outlier within the chosen data set apart from one participant within the exercise group. Moreover, significant outlier should not be present within the dataset. Thus, outlier has been removed and not considered for the test (Kuznetsova et al. 2015).
Multivariate Tests^{a} 

Effect 
Value 
F 
Hypothesis df 
Error df 
Sig. 

health 
Pillai's Trace 
.985 
1160.498^{b} 
2.000 
36.000 
.000 
Wilks' Lambda 
.015 
1160.498^{b} 
2.000 
36.000 
.000 

Hotelling's Trace 
64.472 
1160.498^{b} 
2.000 
36.000 
.000 

Roy's Largest Root 
64.472 
1160.498^{b} 
2.000 
36.000 
.000 

health * Group 
Pillai's Trace 
.963 
472.543^{b} 
2.000 
36.000 
.000 
Wilks' Lambda 
.037 
472.543^{b} 
2.000 
36.000 
.000 

Hotelling's Trace 
26.252 
472.543^{b} 
2.000 
36.000 
.000 

Roy's Largest Root 
26.252 
472.543^{b} 
2.000 
36.000 
.000 

a. Design: Intercept + Group Within Subjects Design: health 

b. Exact statistic 
As per the multivariate test it can be seen that health group has showcased significant outcome with degrees of freedom 2. F Value was high and makes the test aligned with the assumption 5 that argues dependent variable should normally distributed (Parnaudeau et al. 2015).
Mauchly's Test of Sphericity^{a} 

Measure: MEASURE_1 

Within Subjects Effect 
Mauchly's W 
Approx. ChiSquare 
df 
Sig. 
Epsilon^{b} 
GreenhouseGeisser 

health 
.966 
1.234 
2 
.539 
.967 
Mauchly's Test of Sphericity^{a} 

Measure: MEASURE_1 

Within Subjects Effect 
Epsilon 

HuynhFeldt 
Lowerbound 

health 
1.000 
.500 
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.^{a} 

a. Design: Intercept + Group Within Subjects Design: health 

b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of WithinSubjects Effects table. 
Tests of WithinSubjects Effects 

Measure: MEASURE_1 

Source 
Type III Sum of Squares 
df 
Mean Square 
F 

health 
Sphericity Assumed 
3430.762 
2 
1715.381 
1095.397 
GreenhouseGeisser 
3430.762 
1.935 
1773.198 
1095.397 

HuynhFeldt 
3430.762 
2.000 
1715.381 
1095.397 

Lowerbound 
3430.762 
1.000 
3430.762 
1095.397 

health * Group 
Sphericity Assumed 
1557.131 
2 
778.565 
497.172 
GreenhouseGeisser 
1557.131 
1.935 
804.807 
497.172 

HuynhFeldt 
1557.131 
2.000 
778.565 
497.172 

Lowerbound 
1557.131 
1.000 
1557.131 
497.172 

Error(health) 
Sphericity Assumed 
115.883 
74 
1.566 

GreenhouseGeisser 
115.883 
71.587 
1.619 


HuynhFeldt 
115.883 
74.000 
1.566 


Lowerbound 
115.883 
37.000 
3.132 

Homogeneity of variance for each combination of the groups of factors are mandatory for the at least two independent groups. Tests within the subject effect showcase that F value for each health group was high enough and sphericity was justified as the two way mixed anova analysis..
Tests of WithinSubjects Effects 

Measure: MEASURE_1 

Source 
Sig. 

health 
Sphericity Assumed 
.000 
GreenhouseGeisser 
.000 

HuynhFeldt 
.000 

Lowerbound 
.000 

health * Group 
Sphericity Assumed 
.000 
GreenhouseGeisser 
.000 

HuynhFeldt 
.000 

Lowerbound 
.000 

Error(health) 
Sphericity Assumed 

GreenhouseGeisser 


HuynhFeldt 


Lowerbound 

This section of the analysis showcase the within subject effect where the significance of the groups were valid to make the assumption that exercise enhance memory power of the people with slight dementia.
Tests of WithinSubjects Contrasts 

Measure: MEASURE_1 

Source 
health 
Type III Sum of Squares 
df 
Mean Square 
F 
Sig. 
health 
Linear 
2681.899 
1 
2681.899 
2075.396 
.000 
Quadratic 
748.863 
1 
748.863 
407.047 
.000 

health * Group 
Linear 
778.689 
1 
778.689 
602.591 
.000 
Quadratic 
778.442 
1 
778.442 
423.125 
.000 

Error(health) 
Linear 
47.813 
37 
1.292 


Quadratic 
68.071 
37 
1.840 


Within subject contrasts showcase sum of square for the health and health group are good enough to make the chosen mode demonstrate as a good fit model. As the type III sum of squares are much higher and distributed uniformly within the subject they are significant to showcase the model it good fit (Vardigan et al. 2015).
Tests of BetweenSubjects Effects 

Measure: MEASURE_1 Transformed Variable: Average 

Source 
Type III Sum of Squares 
df 
Mean Square 
F 
Sig. 
Intercept 
354812.373 
1 
354812.373 
180397.935 
.000 
Group 
2954.752 
1 
2954.752 
1502.290 
.000 
Error 
72.773 
37 
1.967 


Estimated Marginal Means
1. Group 

Measure: MEASURE_1 

Group 
Mean 
Std. Error 
95% Confidence Interval 

Lower Bound 
Upper Bound 

Exercise 
60.114 
.186 
59.738 
60.490 
No_exercise 
50.060 
.181 
49.693 
50.427 
2. health 

Measure: MEASURE_1 

health 
Mean 
Std. Error 
95% Confidence Interval 

Lower Bound 
Upper Bound 

1 
47.432 
.225 
46.977 
47.887 
2 
58.666 
.238 
58.183 
59.149 
3 
59.163 
.154 
58.851 
59.475 
Estimated marginal means demonstrate health status was better for the group who performed exercise compared to the other who performed conversation with the researcher.
3. Group * health 

Measure: MEASURE_1 

Group 
health 
Mean 
Std. Error 
95% Confidence Interval 

Lower Bound 
Upper Bound 

Exercise 
1 
47.474 
.322 
46.822 
48.125 
2 
67.342 
.341 
66.651 
68.033 

3 
65.526 
.220 
65.080 
65.973 

No_exercise 
1 
47.390 
.313 
46.755 
48.025 
2 
49.990 
.332 
49.316 
50.664 

3 
52.800 
.215 
52.365 
53.235 
Profile Plots
As per the estimated marginal measure variances of the differences between the groups of the between subject factor can be observed. It showcase that people who performed exercise had better mental condition compared to the other.
Leverage was checked using the calculation as follows:
v is the number of predictors in the model and n is the number of data point
Thus, Leverage point is = 2[(2+1)/39]
= .15
From the above analysis it can be seen that physical exercise has enhanced the memory of the participants by a large extent compared to the people who belonged to the nonexercise group. In certain cases as per the test it can be observed that during the midweeks participants scored better than the end result who were in exercise group and reverse can be observed in case of no exercise group. Thus, it can be entailed as per the test that prolonged exercise enhance the memory, however, that do not last for long time. On the other hand, if exercise is not performed and only conversation therapy is utilised for the people with light dementia, it could enhance the memory marginally. All the conditions for the Anova test has been met and it also showcase that data collected by the researcher were justified.
Kuznetsova, A., Christensen, R.H., Bavay, C. and Brockhoff, P.B., 2015. Automated mixed ANOVA modeling of sensory and consumer data. Food Quality and Preference, 40, pp.3138.
Parnaudeau, S., Taylor, K., Bolkan, S.S., Ward, R.D., Balsam, P.D. and Kellendonk, C., 2015. Mediodorsal thalamus hypofunction impairs flexible goaldirected behavior. Biological psychiatry, 77(5), pp.445453.
Vardigan, J.D., Cannon, C.E., Puri, V., Dancho, M., Koser, A., Wittmann, M., Kuduk, S.D., Renger, J.J. and Uslaner, J.M., 2015. Improved cognition without adverse effects: novel M1 muscarinic potentiator compares favorably to donepezil and xanomeline in rhesus monkey. Psychopharmacology, 232(11), pp.18591866.