DATA ANALYSIS DEFINITIONS, TERMS, AND FORMULAS.docx

1. Measures of Central Tendency

Mean (Arithmetic Average)

• Formula: Mean(xˉ)=∑xin\text{Mean} (\bar{x}) = \frac{\sum x_i}{n}
• Example:
  For the dataset [2,4,6,8,10][2, 4, 6, 8, 10], xˉ=2+4+6+8+105=6\bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
• Interpretation:
  The mean value is 6, representing the "average" of the dataset.
  • If the data contains outliers (e.g., [2,4,6,8,100][2, 4, 6, 8, 100]), the mean (2424) may no longer represent the typical value, and the median may be more useful.

Median

• Example:
  For the dataset [2,4,6,8,10][2, 4, 6, 8, 10]:
  Median = 66 (middle value).
  For [2,4,6,8,100][2, 4, 6, 8, 100]:
  Median = 66 (resistant to outliers).
• Interpretation:
  The median is the "middle" value of the dataset and is especially useful for skewed data. If the mean and median differ significantly, the data may be skewed.

Mode

• Example:
  For [1,2,2,3,4][1, 2, 2, 3, 4], the mode = 22.
  For [1,2,3,3,4,4][1, 2, 3, 3, 4, 4], the mode = 33 and 44 (bimodal).
• Interpretation:
  The mode represents the most frequent value(s). Useful for categorical data or datasets with repeating values.

2. Measures of Dispersion

Range

• Formula: Range=Maximum value−Minimum value\text{Range} = \text{Maximum value} - \text{Minimum value}
• Example:
  For [2,4,6,8,10][2, 4, 6, 8, 10], Range = 10−2=810 - 2 = 8.
• Interpretation:
  The range shows the spread between the smallest and largest values. It is sensitive to outliers and does not account for how values are distributed within the range.

Variance

• Formula: Variance(s2)=∑(xi−xˉ)2n−1\text{Variance} (s^2) = \frac{\sum (x_i - \bar{x})^2}{n-1}
• Example:
  For [2,4,6,8,10][2, 4, 6, 8, 10], Mean=6,s2=(2−6)2+(4−6)2+(6−6)2+(8−6)2+(10−6)25−1=10\text{Mean} = 6, \quad s^2 = \frac{(2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2}{5-1} = 10
• Interpretation:
  Variance measures the average squared deviations from the mean. A high variance (e.g., 5050) indicates high variability; a low variance (e.g., 22) indicates less spread.

Standard Deviation

• Formula: Standard Deviation(s)=Variance\text{Standard Deviation} (s) = \sqrt{\text{Variance}}
• Example:
  From the above dataset, s=10≈3.16s = \sqrt{10} \approx 3.16.
• Interpretation:
  Standard deviation measures the average distance of data points from the mean.
  • Low s<0.5s < 0.5: Values are tightly clustered around the mean (low variability).
  • High s>5s > 5: Values are widely spread (high variability).

Interquartile Range (IQR)

• Formula: IQR=Q3−Q1\text{IQR} = Q3 - Q1
• Example:
  For [1,3,5,7,9,11,13][1, 3, 5, 7, 9, 11, 13]:
  Q1=3Q1 = 3, Q3=11Q3 = 11, so IQR=11−3=8IQR = 11 - 3 = 8.
• Interpretation:
  The IQR represents the spread of the middle 50% of the data, resistant to outliers. A small IQR suggests consistency within the central data.

3. Measures of Shape

Skewness

• Example:
  For [1,2,3,4,5,100][1, 2, 3, 4, 5, 100], skewness is positive (tail on the right).
  For [100,5,4,3,2,1][100, 5, 4, 3, 2, 1], skewness is negative (tail on the left).
• Interpretation:
  • Skewness > 0: Right-skewed; mean > median.
  • Skewness < 0: Left-skewed; mean < median.
  • Skewness ≈ 0: Symmetrical distribution.

Kurtosis

• Example:
  For [1,2,3,3,3,4,5][1, 2, 3, 3, 3, 4, 5], kurtosis is positive (leptokurtic).
  For [1,2,3,4,5,6,7][1, 2, 3, 4, 5, 6, 7], kurtosis is negative (platykurtic).
• Interpretation:
  • Kurtosis > 0: Heavier tails (extreme values).
  • Kurtosis < 0: Lighter tails (fewer outliers).
  • Kurtosis = 0: Normal distribution.

4. Visualization

Histogram

• Example:
  A histogram with a long tail on the right indicates positive skew.
• Interpretation:
  The shape of the histogram helps identify modality (unimodal, bimodal) and skewness.

Box Plot

• Example:
  A box plot shows outliers beyond the whiskers.
• Interpretation:
  Useful for identifying spread, center, and outliers.

How to Interpret Results:

• Central Tendency:
  • Mean and median provide insights into the data's center.
  • Compare these to assess symmetry and skewness.
• Dispersion:
  • Small standard deviation (<0.5< 0.5): Data is concentrated around the mean.
  • Large standard deviation (>5> 5): Data is widely spread.
• Shape:
  • Use skewness and kurtosis to infer the type of distribution.

Combining these metrics allows you to summarize your data effectively and draw meaningful conclusions!

Correlation Tests: Understanding the Relationship Between Variables

Correlation tests are statistical methods used to measure the strength and direction of the relationship between two variables. The most common correlation coefficients are:

1. Pearson's Correlation Coefficient (r)

• Formula:
• r = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / √[Σ(xᵢ - x̄)² Σ(yᵢ - ȳ)²]

Where:

• • r: Pearson's correlation coefficient
  • xᵢ: Individual value of variable X
  • x̄: Mean of variable X
  • yᵢ: Individual value of variable Y
  • ȳ: Mean of variable Y
• When to Use:
  • When both variables are continuous and normally distributed.
  • To measure linear relationships.
• Interpretation:
  • r = 1: Perfect positive correlation (as one variable increases, the other increases proportionally)
  • r = -1: Perfect negative correlation (as one variable increases, the other decreases proportionally)
  • r = 0: No linear relationship between the variables

2. Spearman's Rank Correlation Coefficient (ρ)

• Formula:
• ρ = 1 - (6Σd²)/(n(n²-1))

Where:

• • ρ: Spearman's rank correlation coefficient
  • dᵢ: Difference in ranks between corresponding values of X and Y
  • n: Number of pairs of observations
• When to Use:
  • When the data is ordinal or continuous but not normally distributed.
  • To measure monotonic relationships (increasing or decreasing trend).
• Interpretation:
  • Similar to Pearson's correlation coefficient, ranging from -1 to 1.

3. Kendall's Tau (τ)

• Formula:
• τ = (number of concordant pairs - number of discordant pairs) / (number of pairs)

Where:

• • τ: Kendall's tau
  • Concordant pairs: Pairs where the ranking of one variable agrees with the ranking of the other variable
  • Discordant pairs: Pairs where the ranking of one variable disagrees with the ranking of the other variable
• When to Use:
  • When the data is ordinal or continuous but not normally distributed.
  • To measure monotonic relationships.
• Interpretation:
  • Similar to Pearson's and Spearman's correlation coefficients, ranging from -1 to 1.

Interpreting Correlation Coefficients:

• A correlation coefficient of 0.5 indicates a moderate positive relationship between the two variables.
• A value below 0.5, such as 0.3, indicates a weak positive relationship.
• A negative value, such as -0.7, indicates a moderate negative relationship.

Remember: Correlation does not imply causation. A strong correlation between two variables does not necessarily mean one variable causes the other.¹ Other factors may be influencing the relationship.

Understanding these correlation tests and their interpretations can provide valuable insights into the relationships between variables in your data.

Data Analysis Measurements in PLS-SEM Studies

PLS-SEM (Partial Least Squares Structural Equation Modeling) is a powerful statistical technique used to analyze complex relationships between latent variables.¹ It involves two primary models: the measurement model and the structural model.²

Here are the key data analysis measurements used in PLS-SEM:

Measurement Model

1. Reliability:
   • Cronbach's Alpha: Measures the internal consistency of a set of items.³
   • Composite Reliability: Assesses the overall reliability of a construct.⁴
2. Validity:
   • Convergent Validity: Ensures that items measuring a construct are highly correlated with each other.⁵
   • Discriminant Validity: Assures that a construct is distinct from other constructs.⁶ This is often assessed using the Fornell-Larcker criterion or the Heterotrait-Monotrait Ratio of Correlations (HTMT).⁷

Structural Model

1. Path Coefficients:
   • Measure the strength and direction of relationships between latent variables.
   • Significant path coefficients indicate a meaningful relationship.
2. R-square:
   • Indicates the proportion of variance in a dependent variable explained by its independent variables.⁸
   • Higher R-square values suggest a better model fit.
3. Predictive Relevance:
   • Assesses the model's ability to predict future outcomes.⁹
   • Measured using the Stone-Geisser Q² criterion.¹⁰

Additional Considerations:

• Sample Size: A sufficient sample size is crucial for reliable results.¹¹
• Normality: While PLS-SEM is robust to non-normality, it's still important to check for extreme deviations.
• Outliers: Outliers can significantly impact the results.¹² Identify and handle them appropriately.
• Missing Data: Missing data can be addressed using techniques like mean imputation or multiple imputation.¹³

Interpreting Results

• High Reliability and Validity: Indicates that the measurement model is sound.
• Significant Path Coefficients: Suggest meaningful relationships between constructs.
• High R-square: Implies that the model explains a substantial portion of the variance in the dependent variable.¹⁴
• Positive Q² Values: Indicate good predictive relevance.¹⁵

By understanding and applying these data analysis measurements, researchers can effectively evaluate the quality of their PLS-SEM models and draw meaningful conclusions from their findings.

Interpreting Key Values in PLS-SEM

Here's a deeper dive into interpreting key values in PLS-SEM, along with their significance:

Measurement Model

1. Reliability:
   • Cronbach's Alpha: Generally, a value above 0.7 is considered acceptable. It indicates internal consistency among items measuring a construct.
   • Composite Reliability: A value above 0.7 is generally considered acceptable. It represents the overall reliability of a construct.
2. Validity:
   • Convergent Validity:
     • Average Variance Extracted (AVE): A value above 0.5 is generally considered acceptable. It indicates that the construct measures a unique variance.
     • Factor Loadings: Values above 0.7 are generally considered good, indicating strong relationships between items and the construct.
   • Discriminant Validity:
     • Fornell-Larcker Criterion: The square root of the AVE of a construct should be greater than its correlations with other constructs.
     • HTMT (Heterotrait-Monotrait Ratio of Correlations): Values below 0.85 indicate discriminant validity.

Structural Model

1. Path Coefficients:
   • Significance: A significant path coefficient (p-value < 0.05) indicates a statistically significant relationship between two constructs.
   • Effect Size: The magnitude of the path coefficient indicates the strength of the relationship. A larger absolute value indicates a stronger relationship.
2. R-square:
   • A value of 0.2 indicates that 20% of the variance in the dependent variable is explained by the independent variables. Higher values indicate a better model fit.
3. Predictive Relevance:
   • Q-square: A positive Q-square value indicates predictive relevance. A higher value indicates a better predictive power.

Additional Considerations:

• Sample Size: A larger sample size generally leads to more reliable results.
• Normality: While PLS-SEM is robust to non-normality, extreme deviations can affect the results.
• Outliers: Outliers can significantly impact the results. Identify and handle them appropriately.
• Missing Data: Missing data can be addressed using appropriate techniques.

By understanding and interpreting these values, researchers can assess the quality of their PLS-SEM models and draw meaningful conclusions from their findings.

Would you like to delve deeper into a specific aspect of PLS-SEM or discuss a particular research scenario?