Prac 1
No major gender difference in Lo D, Hi F ondition but in Hi D, Hi F condition females show slightly higher median Fright and larger variablity with few low-end outliers in both the groups.
H0: Homogenity of variances against H1: Not H0
As t=-2.267, p=0.029, Cohen's d=-0.6841
We reject H0 at alpha level of significance
Cohen's d represents medium to large effect size between differences of group.
Assumption Check:
1.W=0.9734,p=0.396
Since p>0.05 then assumption of normality holds
2.F=2.362,p=0.132
Since p>0.05 then assumption of normality holds
Prac 2
q.1 (a)
Interpretation:
In the classical ANOVA, both Antidepressant and Cognitive_Therapy have statistically significant effects on After_Treatment_Score (p < .001). Their interaction is not statistically significant (p = .506).
After adding Baseline_Score as a covariate in ANCOVA, the effects of Antidepressant and Cognitive_Therapy remain statistically significant (p < .001), while their interaction also becomes statistically significant (p = .004).
Thus, after controlling for Baseline_Score, the p-values of the treatment effects change and the interaction effect becomes significant.
Interpretation:
(b)
Baseline_Score is a statistically significant covariate (F = 135.5, p < .001). Therefore, the initial depression score has a significant effect on the After_Treatment_Score.
steps:
upload depression dataset->select classical anova->put after treatment score in dependent variable and anti depressant and cognitive therapy in fixed factors->select classic ancova and put baselene score in covariates
q.2
(a)Interpretation:
All four multivariate tests (Pillai's Trace, Wilks' Lambda, Hotelling-Lawley Trace, and Roy's Largest Root) show a statistically significant effect of Group on the combined dependent variables Actions and Thoughts (p < .001).
Thus, all multivariate tests agree that the therapy Group has a significant multivariate effect.
(b) Interpretation:
The separate univariate ANOVA results show that Group has a statistically significant effect on both Actions (F = 68.22, p < .001) and Thoughts (F = 44.10, p < .001).
The multivariate MANOVA and individual ANOVA results both indicate a significant effect of Group. MANOVA examines the combined dependent variables, while univariate ANOVA examines each dependent variable separately.
q.3
(a) Interpretation:
The scatterplot shows a positive relationship between population size (Pop) and crime rate (Crime). Thus, states with larger populations generally tend to have higher observed crime rates.
(b) Interpretation:
A positive correlation between police expenditure (Pol) and crime does not imply that increased police funding causes more crime. Correlation does not establish causation. Other factors, such as population size and socioeconomic conditions, may affect both police expenditure and crime. Also, higher crime may lead to increased police spending.
q.4
(a) Interpretation:
The multiple linear regression model explains 64.9% of the total variation in state crime rates (R² = 0.649). The Adjusted R² is 0.539, indicating that 53.9% of the variation is explained after adjusting for the number of predictors in the model. The overall regression model is statistically significant (F = 5.885, p < .001).
(b) Interpretation:
The statistically significant predictors of Crime are U1 (p = .012), U2 (p = .008), Wealth (p = .009), Ineq (p = .012), and Ed (p = .010).
Among these significant predictors, U1 has a negative relationship with Crime (B = -12830). U2, Wealth, Ineq and Ed have positive relationships with Crime.
Prac 3:
q.1
MANOVA Q1
a:H0: There is no significant difference between CBT and BT v/s H1: Not H0
b. pillai trace value=0.8374
transformed F=18.02
p=0.002
c.T^=41.18
d. We reject H0 as H0 is highly significant and they produce significantly different multivariate treatment profiles
q.2
a. H0: Sigma1=Sigma2 vs H1: not H0
q.3
c. Chi-Square=3.856, df=3, p=0.277,f=18.02
d. we do not reject H0
Prac 4
Question 1
(a) Hypotheses for Bartlett's Test of Sphericity:
Null Hypothesis (H0): The population correlation matrix is an identity matrix (R=I). That is, the 6 items are mutually uncorrelated in the population.
Alternative Hypothesis (H1): The population correlation matrix is not an identity matrix (R≠I) . Significant inter-correlations exist among the items.
Implication: Failing to reject the null hypothesis (p > 0.05) indicates that the variables share little to no common variance. This implies that the data matrix is unsuitable for factor analysis, as there are no underlying latent constructs driving the relationships between the items.
(b) Sampling Adequacy & Sphericity Check Interpretation
Measure of Sampling Adequacy (MSA):
The Overall Kaiser-Meyer-Olkin (KMO) index is 0.914, which falls in the "marvelous" range (KMO>=0.90).
Individual item MSAs range from 0.871 (Anxiety_3) to 0.951 (Focus_2), all well above the minimum standard threshold of 0.50. This confirms excellent factorability.
Bartlett's Test of Sphericity: Χ215 = 157.084, p < .001. Since p < .001, H0 is rejected, confirming significant inter-variable correlations.
(c) Factor Extraction & Scree Plot Justification
Parallel Analysis Engine: Parallel analysis identifies 1 factor, as only the first eigenvalue ( λ1= 5.367) exceeds the random data baseline (~1.8).
Principal Axis Factoring: The primary factor accounts for 89.45% of the total variance, whereas Factor 2 accounts for only 1.94% (λ2= 0.116).
Scree Plot Interpretation: The plot shows a clear elbow after Factor 1. Only the first point remains above the parallel analysis threshold line, justifying a single-factor structure for the underlying data. When manually extracting a 2-factor solution, Factor 2 contributes minimal additional variance.
(Note: Retention threshold set to > 0.40.)
Substantive Psychological Construct Labels
Factor 1: Cognitive Focus / Attention Control
Key Marker: Focus_3 (loading = 0.944)
Interpretation: Measures task engagement, concentration, and mental clarity.
Factor 2: Psychological Anxiety / Distress
Key Marker: Anxiety_3 (loading = 0.942)
Interpretation: Measures affective and somatic symptoms of anxiety.
(a) PATH DIAGRAM
The proposed two-factor measurement model consists of two latent constructs:
1. Anxiety → Anxiety_1, Anxiety_2, Anxiety_3
2. Focus → Focus_1, Focus_2, Focus_3
Each observed variable has its own error term (δ1–δ6). Anxiety and Focus are allowed to covary, and no cross-loadings are allowed.
SEM notation:
Anxiety =~ Anxiety_1 + Anxiety_2 + Anxiety_3
Focus =~ Focus_1 + Focus_2 + Focus_3
Anxiety ~~ Focus
CFA Model*
(c) MODEL FIT RESULTS
Chi-square (χ²) = 5.018
Degrees of freedom (df) = 8
p-value = 0.756
CFI = 1.000
RMSEA = 0.000
90% CI for RMSEA = 0.000 to 0.1846
Interpretation:
The chi-square test is non-significant (p = 0.756 > 0.05), indicating no significant difference between the observed and model-implied covariance matrices. CFI = 1.000 exceeds the 0.95 criterion and RMSEA = 0.000 is below the 0.06 criterion. Therefore, the strict two-factor model shows excellent overall model fit.
(d) INTERPRETATION
All indicator factor loadings are statistically significant (p < .001), indicating that the observed items significantly measure their respective factors.
The estimated covariance between Anxiety and Focus is -2.240 (SE = 0.745, z = -3.004, p = .003), indicating a statistically significant negative relationship.
The standardized correlation between Anxiety and Focus is approximately -0.99. Thus, the two factors share approximately 97.8% of their variance (r² = 0.978).
Although the strict two-factor CFA model shows excellent fit (CFI = 1.000, RMSEA = 0.000), the extremely high inverse correlation indicates very low discriminant validity between Anxiety and Focus.
Conclusion:
The two-factor model fits the data well, but Anxiety and Focus are extremely strongly and negatively related and are not clearly distinguishable as separate constructs in this dataset. The results therefore suggest that the two factors may collapse into a single bipolar distress construct.
prac 6
MAHALANOBIS DISTANCE
(a) Formula:
D² = (xᵢ - x̄)ᵀ S⁻¹ (xᵢ - x̄)
Where:
xᵢ = Observation vector
x̄ = Mean vector
S⁻¹ = Inverse of covariance matrix
T = Transpose
(b) Distribution and Degrees of Freedom:
Mahalanobis D² approximately follows a Chi-square (χ²) distribution.
Number of variables (p) = 4
Therefore,
df = 4
(c) Procedure in JASP:
{
1. Open the Iris dataset in JASP.
2. Create a new variable ID using the formula 1:150.
3. Go to Regression → Linear Regression.
4. Put ID in the Dependent Variable.
5. Put Sepal.Length, Sepal.Width, Petal.Length and Petal.Width in Covariates.
6. Open Diagnostics.
7. Select Mahalanobis distance.
8. Check the Case Diagnostics table.}
(d) Cut-off and Conclusion:
For df = 4 and α = 0.001:
χ² critical value = 18.467
Maximum mahalanobis distance D² = 13.10
If D² > 18.467, the observation is considered a possible multivariate outlier.
The maximum Mahalanobis distance in the Iris dataset is below 18.467.
Conclusion:
There are no significant multivariate outliers in the Iris dataset at α = 0.001.
(e) Multivariate Outlier:
A multivariate outlier may have normal values for each variable individually but may be unusual when all variables are considered together.
Mahalanobis distance detects such unusual combinations by considering the relationships between variables.
Q2) (A) Clustering Setup:
Four morphological variables were used to form 3 K-Means clusters.
(B) Cluster Sizes:
Cluster 1 = 53, Cluster 2 = 50, Cluster 3 = 47.
The clusters are fairly balanced.
(C) Clustering Performance:
R² = 0.767, showing that 76.7% of the variation is explained
by the 3-cluster solution.
Silhouette = 0.460, indicating moderate cluster separation.
(D) Cluster Structure:
The clusters mainly differ in Petal.Length and Petal.Width.
Cluster 2 has lower petal values, while Cluster 3 has higher values.
(E) Validation:
Setosa: 50/50 correctly classified (100%).
Versicolor: 39/50 correctly classified.
Virginica: 36/50 correctly classified.
Some misclassification occurs between Versicolor and Virginica
due to similarity in their morphological features.
χ² = 187.6, df = 4, p < .001, showing a significant association
between species and cluster membership.