Psychological Statistics Notes PDF

Psychological Statistics 1.0 What is Statistics? 1.2 Inferential Statistics Psychological statistics involves the application of statistical methods to Purpose: Inferential statistics encompass a range of techni...

Psychological Statistics 1.0 What is Statistics? 1.2 Inferential Statistics Psychological statistics involves the application of statistical methods to Purpose: Inferential statistics encompass a range of techniques that analyze data in psychological research. It helps researchers collect, allow researchers to draw conclusions about populations from sample summarize, and interpret data to draw meaningful conclusions about data, test hypotheses, and explore relationships between variables behavior, cognition, and emotions. through correlation and regression analysis. 1.1 Descriptive Statistics Hypothesis Testing Purpose: Descriptive statistics involve summarizing and organizing data Null Hypothesis (H₀): States that there is no effect or no difference in so that it can be easily understood. These statistics help to describe, the population. It is the default assumption in hypothesis testing. show, or summarize data in a meaningful way. Alternative Hypothesis (H₁): The hypothesis that suggests there is an Note: Descriptive statistics do not make predictions or conclusions beyond effect or a difference in the population. the data itself. P-value: The probability of observing a result as extreme as the one in your data, assuming the null hypothesis is true. A p-value of less than Measures of Central Tendency 0.05 usually indicates statistical significance, leading to the rejection of the null hypothesis. These represent the center or typical value in a data set. Type I Error (α): Rejecting the null hypothesis when it is actually true Mean: The sum of all values divided by the total number of values. Most (false positive). commonly used measure of central tendency. Type II Error (β): Failing to reject the null hypothesis when the Median: The middle value in an ordered dataset; for even-sized alternative hypothesis is true (false negative). datasets, it's the average of the two middle values. Mode: The most frequently occurring value in the data set. A data set Confidence Intervals can have one mode, more than one mode, or no mode. Definition: A range of values derived from the sample data that is Measures of Dispersion likely to contain the true population parameter. Purpose: Instead of providing a single estimate, it offers a range These describe how spread out or clustered the data points are in a data within which the population parameter is expected to fall. set, helping to understand variability Common Level: 95% confidence interval, meaning that there is a 95% Range: The difference between the highest and lowest values in the data probability that the interval contains the true population parameter. set. It provides a simple measure of dispersion but can be influenced by outliers. Descriptive Vs Inferential Variance: The average of the squared differences from the mean. It gives a sense of how far data points are from the mean, with larger values indicating greater variability. Aspect Descriptive Inferential Standard Deviation: The square root of variance, representing Summarizes data from a Makes predictions or inferences dispersion in the same units as the data. A higher standard deviation Purpose sample about a population indicates that data points are more spread out from the mean. Interquartile Range (IQR): The difference between the first (Q1) and Data Focuses on the sample or data Applies findings from the third quartiles (Q3). It shows the range within which the middle 50% of data Scope at hand sample to a broader population falls, reducing the impact of outliers. Describes what has been Tests hypotheses or makes Use observed generalizations Frequency Distributions Mean, median, mode, standard T-tests, ANOVA, regression, Examples deviation, range confidence intervals These summarize how often each value or range of values occurs in the data set. Definition: A frequency distribution is a table or graph that shows the 2.0 Uses of Statistics number of occurrences (frequency) of each unique value or set of values within a data set. Conducting a survey of a small group of people to estimate the voting preferences of an entire city, or analyzing the relationship between Types: education level and income to predict future earnings. Simple Frequency Distribution: Lists each value along with its Calculating the average age of students in a class, finding the most frequency. common eye color, or creating a pie chart showing the percentage of Grouped Frequency Distribution: Groups data into intervals or different car brands in a parking lot. ranges and shows the frequency of each interval. Histograms: A graphical representation of a frequency distribution where the x-axis represents the data values or intervals, and the y-axis represents the frequency of occurrences. she 🐧 | Review Well~ Course Enrichment I Page 1  3.0 Population vs Sample Interval Scale 3.1 Sample Definition: Numerical data with equal intervals between values, but no true zero point. (How far apart are the data from each other) A sample is a subset of individuals selected from the population.. Examples: In the best case, the sample will be representative of the population. Temperature (Celcius, Fahrenheit) That is, the characteristics of the individuals in the sample will Years (1990, 2000, 2010) mirror those in the population. Statistical Analysis: Frequency counts, modes, medians, means, standard deviation, correlation, and t-tests. 3.2 Population Ratio Scale A population is the entire set of individuals that we are interested in studying. Definition: Numerical data with equal intervals between values and a Thus, it is usually not feasible to collect data from the entire true zero point. population. Examples: Weight (kilograms, pounds) 4.0 Variable Length (meters, feet) Statistical Analysis: All statistical analyses can be applied to ratio Variable: Any characteristic, number, or quantity that can be measured or data, including geometric mean, coefficient of variation, and ANOVA. counted and may vary across individuals or observations in a dataset. Qualitative Variable: A non-numerically valued variable. Equal True Scale Order Statistical Analysis Quantitative Variable: A numerically valued variable. Intervals Zero Independent Variable: The variable that is manipulated or controlled Frequency counts, mode, in an experiment to see its effect on another variable. Nominal No No No chi-square tests Dependent Variable: The variable that is measured in an experiment and is expected to change in response to the independent variable. Frequency counts, modes, Ordinal Yes No No Control Variable: Variables that are kept constant to ensure that the medians, non-parametric tests relationship between the independent and dependent variables is not Frequency counts, modes, affected by outside influences. Interval Yes Yes No medians, means, standard deviation, correlation, t-tests 5.0 Data Ratio Yes Yes Yes All statistical analyses Qualitative Data: Categorical Quantitative Data: 6.0 Frequency Distribution Data that consist of numbers obtained from counts or measurements. Quantitative data can further be described by distinguishing between Frequency distribution is a descriptive statistical tool that organizes discrete and continuous types. data into categories or intervals, showing how often each value or range of values occurs in a dataset. 5.1 Kinds of Data Purpose: Visualize Data: Provides a clear, structured way to display the Nominal Scale distribution of data. Identify Patterns: Helps in spotting trends, such as most common or Definition: Categorical data with no inherent order. rare occurrences. Examples: Simplify Data: Organizes large datasets into manageable summaries. Colors ( red, blue, green) Gender (male, female, other) 6.1 Key Elements Statistical Analysis: Frequency counts, modes, and chi-square tests. Categories or Intervals: The data is grouped into distinct intervals or Ordinal Scale categories. These can be individual values or ranges (e.g., 1-10, 11-20). Frequency: The number of occurrences of each category or interval in the dataset. Definition: Categorical data with a natural order. Relative Frequency: This expresses frequency as a proportion or Examples: percentage of the total number of observations. Educational levels (elementary, high school, college) Cumulative Frequency: The running total of frequencies up to a Likert scale (disagree, neutral, agree) certain category or interval. Statistical Analysis: Frequency counts, modes, medians, and non-parametric tests (e.g., Mann-Whitney U test) she 🐧 | Review Well~ Course Enrichment I Page 2  7.0 Graph Analysis Different types of graphs help visualize data in unique ways, allowing for clearer understanding and analysis of various types of data. 7.1 Types of Graphs and Their Uses Line Graph Description: A graph that uses points connected by lines to show changes over time. Uses: Displaying changes over a period (time-series data). Tracking patterns or trends. Histogram Description: Similar to a bar graph, but the bars represent ranges or intervals of continuous data, and the bars touch each other. Uses: Displaying the distribution of continuous data. Identifying the shape of the data distribution (normal, skewed). Bar Graph Description: Uses rectangular bars to represent different categories. The length or height of each bar corresponds to the frequency or value of the category. Uses: Displaying changes over a period (time-series data). Tracking patterns or trends. Scatter Plot Description: A graph that uses points to represent the values of two different variables on two axes. Each point shows the relationship between the two variables. Uses: Examining relationships or correlations between variables. Identifying patterns or clusters. Pie Chart Description: A circular graph divided into slices where each slice represents a proportion of the whole. Uses: Representing parts of a whole. Displaying percentage or proportional data. she 🐧 | Review Well~ Course Enrichment I Page 3  Interpretation: A standard deviation of 1 means that the data points 7.2 How to Understand Graphs are, on average, 1 unit away from the mean, and so on. Trends: Look for patterns or changes in the data.. 9.0 Skewness Extremes: Identify the highest and lowest points. Outliers: Notice any unusual data points. Comparisons: Compare different categories or time periods. Skewness measures the asymmetry or distortion of a probability distribution, showing how it deviates from a symmetric distribution like 8.0 Normal Distribution Curve the normal distribution. A normal distribution curve, often referred to as a bell curve, is a symmetrical probability distribution that shows how data is distributed around a mean value. Positive skewness means that most of the data values are low, but there are a few higher values pulling the tail of the distribution to the right. Negative skewness means that most of the data values are high, but there are a few lower values pulling the tail of the distribution to the left. Why is Skewness Important? Skewness can affect the accuracy of statistical analyses that assume a normal distribution. 8.1 Properties of an NDC Skewness can provide insights into the underlying distribution of data, which can be helpful for decision-making and risk assessment. Symmetry: The curve is symmetrical around the mean, meaning that Skewness can be used to identify outliers and other unusual data points. half of the data points are on each side. Bell shape: The curve resembles a bell, with the highest point at the How to measure skewness? mean. Central Limit Theorem: Mean, median, and mode are equal: In a Visual Inspection: A histogram or a density plot can be used to visually normal distribution, these three are the same. assess skewness. Standard Deviation: The shape of the bell is different determined by Skewness Coefficient: The skewness coefficient is a numerical the standard deviation. A smaller standard deviation results in a narrower, measure of skewness. It can be calculated using the mean, median, and taller curve, while a larger standard deviation results in a wider, flatter mode of the data. curve. 10.0 Kurtosis 8.2 Central Tendencies These represent the center or typical value in a data set. Kurtosis: is a statistical measure that describes the "tailedness" or "peakedness" of a probability distribution. It reveals how much data is Mean: The sum of all values divided by the total number of values. Most commonly used measure of central tendency. concentrated in the tails versus the center, offering insights into the Median: The middle value in an ordered dataset; for even-sized distribution's shape. Understanding kurtosis helps assess the likelihood of datasets, it's the average of the two middle values. extreme values, aiding decision-making in fields like finance, statistics, and Mode: The most frequently occurring value in the data set. A data set data science. can have one mode (unimodal), more than one mode(multimodal), or no mode. Why is Kurtosis Important? 8.3 Standard Deviation Risk Assessment: In finance, kurtosis assesses risk. A leptokurtic Standard Deviation: A statistical measure that shows how spread out distribution signals higher risk due to potential extreme events like the data points are from the mean. market fluctuations or economic downturns. Low Standard Deviation: Indicates that the data points are clustered Data Analysis: Understanding kurtosis helps select suitable statistical closely around the mean. models and accurately interpret data. High Standard Deviation: Indicates that the data points are spread Investment Decisions: Investors consider kurtosis to identify out over a wider range. potential risks and rewards when making investment choices. she 🐧 | Review Well~ Course Enrichment I Page 4  Kurtosis Graphs Deciles Deciles: Statistical values that divide a dataset into ten equal parts. Each decile represents 10% of the data, allowing for a more granular analysis than quartiles (which divide data into four parts) or percentiles (which divide data into 100 parts). How Deciles Work: Order the Data: Arrange the data points in ascending or descending order. Divide into Tenths: Identify the points that divide the data into ten Mesokurtic (Normal Distribution): The mesokurtic distribution has a equal parts. kurtosis of 3, with a moderate peak and data evenly around the mean. Most values cluster near the average, leading to a balanced number of Uses of Deciles: outliers. Statistics: Analyzing data distribution and identifying outliers. Leptokurtic: The leptokurtic distribution has a kurtosis greater than 3, Finance: Assessing investment performance and risk. characterized by a sharp peak and heavy tails. This indicates a higher Education: Evaluating student performance and comparing schools. likelihood of extreme values, making it riskier. Platykurtic: The platykurtic distribution has a kurtosis less than 3. It Percentiles features a flatter peak and light tails, suggesting the data is more evenly spread with fewer outliers. Percentiles: Statistical measure that divides a dataset into 100 equal parts. Each percentile represents a specific point within the data distribution, 11.0 Variability indicating the percentage of values that fall below it. Variability: In statistics refers to how spread out or dispersed data points How Percentiles Work: are within a dataset. It's a measure that helps us understand how much Data Ordering: Arrange the data points in ascending order. the data varies from its average or central tendency. Division: Identify the points that divide the data into 100 equal parts. Percentile Calculation: The percentile value separates a specified 11.1 Why is Variability Important? percentage of data from the rest. Examples: Sampling: The process of selecting a subset of individuals or elements 25th Percentile: This is also known as the first quartile (Q1). It from a larger population to make inferences about that population. The indicates that 25% of the data points are below it. goal is to obtain a representative sample that accurately represents the 50th Percentile: This is also known as the median. It represents the characteristics of the population. middle value in the dataset, meaning 50% of the data points are below it. 75th Percentile: This is also known as the third quartile (Q3). It 11.2 Measures of Position indicates that 75% of the data points are below it. Quartiles 12.0 Inferential Statistics Three Main Quartiles: 12.1 Sampling First Quartile (Q1): This is the median of the lower half of the data. It represents the 25th percentile, meaning 25% of the data lies below this Sampling: The process of selecting a subset of individuals or elements value. from a larger population to make inferences about that population. The Second Quartile (Q2): This is also known as the median. It divides goal is to obtain a representative sample that accurately represents the the data into two equal halves. characteristics of the population. Third Quartile (Q3): This is the median of the upper half of the data. It represents the 75th percentile, meaning 75% of the data lies below this 12.2 Qualities of a Good Sample value. Why Use Quartiles? Representativeness: The sample should reflect the characteristics of the Understanding Data Distribution: Quartiles help visualize how data population being studied, including demographics and key variables. is spread out. Accuracy: The sample should provide accurate information about the Identifying Outliers: By comparing data points to quartiles, you can population. spot outliers or extreme values. Precision: The sample should provide a precise estimate of the Calculating the Interquartile Range (IQR): The IQR, which is the population being studied. difference between Q3 and Q1, is a measure of variability or spread. It's Reliability: The sample should produce consistent results if the study were to be repeated. often used to identify outliers. Validity: The sample should measure what it is intended to measure, ensuring that the findings can be accurately generalized to the population. she 🐧 | Review Well~ Course Enrichment I Page 5  12.3 Sampling Techniques 13.0 Parametric Tests Probability Sampling uses random selection to enhance Parametric Tests: Statistical tests that make assumptions about the representativeness and allow for generalizations about the population. underlying distribution of the data. These assumptions typically involve Non-Probability Sampling involves non-random selection based on the data being normally distributed and having equal variances. convenience or judgment, which can introduce bias and limit Normality: The data is assumed to follow a normal distribution. generalizability. Homoscedasticity: The variance of the data is assumed to be equal across groups. Probability Sampling 13.1 When to Use Parametric Tests Simple Random Sampling: Every individual has an equal chance of being selected, often using random number generators. Normal Distribution: Data follows a normal distribution Stratified Sampling: The population is divided into subgroups (strata), Interval or Ratio Data: Data is measured at the interval or ratio level. and random samples are taken from each stratum to ensure representation Homogeneity of Variance: Variance among groups is approximately of key characteristics. equal (homoscedasticity). Systematic Sampling: Individuals are selected at regular intervals Independence: Samples are independent of each other. from a sorted list (e.g., every 10th person). Sample Size: Ideally, use larger samples to meet normality Cluster Sampling: The population is divided into clusters, and entire assumptions. clusters are randomly selected for study. (What makes one group is what also makes other groups, like geographical location) 13.2 Advantages and Disadvantages Multi-Stage Sampling: This technique combines multiple sampling methods. Initially, clusters are randomly selected, and then a random Advantages: sample is taken from each selected cluster, allowing for a more Statistical Power: Greater power allows for detecting differences with manageable and representative sample from large populations. smaller sample sizes. Normality Assumption: More accurate results when data follow a Non-probability Sampling normal distribution. Detailed Results: Provides more information, such as means and Convenience Sampling: Samples are drawn from individuals who are variances. readily available, which can result in bias and reduced representativeness. Widespread Use: Common methods (e.g., T-tests, ANOVA) are Judgmental Sampling: The researcher selects individuals based on well-accepted. their judgment about who will be most informative. Disadvantages: Snowball Sampling: Existing study subjects recruit future subjects from Assumption Dependence: Results can be inaccurate if assumptions among their acquaintances, useful in hard-to-reach populations. (e.g., normality, homogeneity of variance) are violated. Quota Sampling: The researcher ensures equal representation of Sensitivity to Outliers: Outliers can skew results and affect validity. specific characteristics by setting quotas for different groups within the Data Type Limitations: Requires interval or ratio data, unsuitable for population. ordinal or nominal data. Purposive Sampling: The researcher selects individuals purposefully Less Robust: Small sample sizes or non-conforming data can lead to based on specific criteria or characteristics relevant to the study, ensuring unreliable results. that the sample is aligned with the research objectives. 13.3 Common Parametric Tests 12.4 Sampling Error Sampling error is the difference between the sample statistic and the actual T-Test population parameter. It occurs due to: Random Variability: Even with good sampling techniques, samples T-test: Used to determine if there is a significant difference between the may not perfectly represent the population, leading to variations. means of two groups. Sample Size: Smaller samples tend to have larger sampling errors due to Commonly used in hypothesis testing to assess whether a process or increased variability. treatment has a meaningful effect on a population. Bias: Non-random sampling methods can lead to systematic errors, Types of T-Tests: affecting the accuracy of inferences. One-Sample T-test: Compares the mean of a single sample to a known Reducing Sampling Error population mean. For example, you might use this to determine if the Increase the sample size to enhance representativeness and reduce average height of students in a class is significantly different from the variability. national average. Use random sampling techniques to minimize bias. Independent Samples T-test: Compares the means of two independent groups. For example, you could use this to compare the test scores of students who received a new teaching method to those who received the traditional method. she 🐧 | Review Well~ Course Enrichment I Page 6  Paired Samples T-test: Compares the mean of two paired or dependent When to Use ANOVA? samples. This is often used when the same individuals are measured twice, Comparing Multiple Groups: When you want to determine if there are such as before and after a treatment or intervention. significant differences between the means of more than two groups. Testing the Effectiveness of Treatments: To evaluate the efficacy of Interpretation of T-Test: different treatments or interventions. Rejecting the Null Hypothesis: This indicates that there is a Analyzing Experimental Data: To assess the impact of different factors statistically significant difference between the means. or conditions on a dependent variable. Failing to Reject the Null Hypothesis: This suggests that there is Understanding Relationships Between Variables: To explore the not enough evidence to conclude a significant difference between the means. relationships between a categorical independent variable and a quantitative dependent variable. Example of T-Test: A teacher wants to compare the average test scores of students in two Example of ANOVA: different classes: Class A and Class B. She hypothesizes that there is no A researcher wants to compare the average test scores of students from significant difference in the average test scores between the two classes. three different schools. Using ANOVA, they can determine if there is a T-statistic: 0.54 | P-value: 0.61 | Degrees of freedom: 8 significant difference in the test scores among the schools. Since the p-value (0.61) is greater than the typical significance level (0.05), we fail to reject the null hypothesis. This means that there is not enough Factorial Notation evidence to conclude that there is a significant difference in the average test scores between Class A and Class B. Factorial Notation: Is a shorthand way of representing the number of levels for each factor (independent variable) in an experiment. Each factor is Analysis of Variance (ANOVA) denoted by a number, and the levels of each factor are separated by a multiplication sign. ANOVA: Used to determine whether there are significant differences Why is Factorial Notation Used? between the means of three or more groups. Clarity: It provides a concise and clear representation of the experimental design. How Does ANOVA Work? ANOVA divides the total variance in the data into two components: Efficiency: It simplifies the description of complex designs with multiple factors and levels. Standard: It is a widely used convention in statistical analysis, making it easier to communicate and understand research findings. Example? 2x3 Factorial Design: This means there are two factors. Factor A has 2 levels, and Factor B has 3 levels. 3x2x4 Factorial Design: This means there are three factors. Factor A has 3 levels, and Factor B has 2 levels, and Factor C has 4 Levels. Levene’s Test Between-Group Variance: The variation among the means of different groups. Levene’s Test: Is used to determine if the variances of multiple samples Within-Group Variance: The variation within each group. are equal. This assumption of equal variances, often referred to as F-Statistic: The ANOVA calculates an F-statistic, which is the ratio of homoscedasticity, is a crucial prerequisite for many statistical analyses, between-group variances to within-group variance. such as ANOVA and t-tests. Hypothesis Testing: If the F-statistic is significantly larger than 1, it suggests that the between-group variances is substantially greater than Why is it Important? the within-group variance, indicating that there are significant differences Assumption of Equal Variances: Many statistical tests, like ANOVA between the group means. and t-tests, assume that the variances of the groups being compared are equal. If this assumption is violated, the results of these tests can be Types of ANOVA: unreliable. One-way ANOVA: Compares the means of three or more groups based Robustness: Levene’s test is relatively robust to violations of normality, on a single factor or independent variable. making it a suitable choice for many datasets. Two-way ANOVA: Compares the means of groups based on two factors or independent variables. Repeated Measures ANOVA: Used when the same individuals are Fisher’s Score measured multiple times under different conditions or at different time points. Fisher’s Score: Is a measure of how discriminative a feature is in Mixed Model ANOVA: Combines features of one-way and repeated predicting a binary outcome. measures ANOVA, involving both between-subject and within-subject Fisher score is often used in feature selection for classification problems. factors. It can help to identify the most important features and reduce the dimensionality of the data, which can improve performance of a classification model. she 🐧 | Review Well~ Course Enrichment I Page 7 Advantages: Tukey Post-hoc It is a simple and computationally efficient method. It is good measures of feature discriminativeness. It can be used with both numerical and categorical features. Tukey's Honestly Significant Difference (HSD) Test: Is a statistical procedure used to determine which specific groups differ significantly from Disadvantages: one another after a significant ANOVA result. It assumes that the features are normally distributed. It can be sensitive to outliers. When to Use? It may not be as effective as other methods for highly correlated features. After a significant ANOVA: If your ANOVA indicates that there are significant differences among your groups, Tukey's HSD can help you pinpoint which groups are different. Welch’s t-test Comparing all pairs: If you want to examine every possible combination of group means, Tukey's HSD is a good choice. Welch’s t-test: Is used to compare the means of two independent samples when the variances of the two populations are not assumed to be equal. Key Points: Controls family-wise error rate: Tukey’s HSD helps to control the When to Use Welch’s t-test? overall probability of making a Type I error when conducting multiple Unequal Variances: When the standard deviations or variances of the comparisons. two groups are significantly different. Assumes Equal Variances: It assumes that the variances of the groups Independent Samples: When the data points in one group are are equal. independent of the data points in the other group. Robust to Violations of Normality: While it’s generally robust to Example Use Cases: violations of normality, especially with larger sample sizes, it’s still best to Comparing the average test scores of two different classes. check for normality. Comparing the average salaries of two different professions. Evaluating the effectiveness of two different marketing campaigns. Correlation Analysis Sphericity Definition: Assesses the strength and direction of the relationship between two variables. Does not imply causation! Sphericity: Assumes that the variances of the differences between all pairs Purpose: Helps researchers understand whether and how strongly pairs of repeated measures are equal. When this assumption is violated, the F-test of variables are related, guiding further analysis. in repeated measures ANOVA becomes inflated, leading to an increased Correlation Coefficient (r): A statistical measure that indicates the likelihood of incorrectly rejecting the null hypothesis. extent to which two variables fluctuate together. Positive Correlation (0 to +1): As one variable increases, the other variable also tends to increase. Correction Formula for Sphericity Violation Negative Correlation (-1 to 0): As one variable increases, the other variable tends to decrease. Greenhouse-Geisser Correction: Adjusts degrees of freedom when the No Correlation (0): There is no relationship between the variables. assumption of sphericity (equal variances of differences between repeated measures) is violated in repeated measures ANOVA. Conservative approach: It makes a larger adjustment to the degrees of Regression Analysis freedom, which reduces the risk of Type I error but can make it harder to find significance. Definition: Used to predict the value of a dependent variable based on The Greenhouse-Geisser correction calculates an estimate of one or more independent variables. Implies causation! sphericity, denoted by ε (epsilon). This estimate ranges from 0 to 1. If ε Purpose: Helps in understanding the relationship between variables and = 1, sphericity is assumed to hold. If ε < 1, there is a violation of sphericity. in making predictions. It provides insight into how the dependent variable changes with variations in the independent variable(s). Huynh-Feldt Correction: A statistical adjustment used in repeated Types: measures ANOVA to address violations of the sphericity assumption. Simple Linear Regression: Examines the relationship between two The Huynh-Feldt correction is generally recommended when the variables by fitting a linear equation to the observed data. It determines estimated epsilon value is greater than 0.75. If epsilon is less than or equal to how much the dependent variable changes when the independent variable 0.75, the Greenhouse-Geisser correction is often preferred as it tends to be changes by one unit. more conservative. Multiple Regression: Involves two or more independent variables to Aggressive: It makes a smaller adjustment to the degrees of freedom, so it's predict the dependent variable, allowing for more complex relationships more likely to detect significant effects compared to Greenhouse-Geisser. and interactions. When to Use? Greenhouse-Geisser is preferred when there's a strong violation of sphericity, as it more rigorously prevents Type I error. Huynh-Feldt is better when there's a mild or moderate violation of sphericity, as it strikes a balance between controlling for error and retaining statistical power. she 🐧 | Review Well~ Course Enrichment I Page 8  Disadvantages: Summary Less Statistical Power: Generally less powerful than parametric tests, requiring larger samples to detect differences. Test When to Use? Key Points Less Detailed Information: Provides less information about the data, Independent Samples Compare means of two such as means or variances. Normal data, equal variances. T-Test independent groups. Limited Applicability: May not be suitable for certain analyses Compare means of two related Related data, normal where parametric tests are more appropriate. Paired Samples T-Test groups (e.g., before/after). distribution. Compare means of three or One-Way ANOVA Normal data, equal variances. 14.3 Common Non-parametric Tests more independent groups. Compare effects of 2+ Used in designs like 2x2, Factorial ANOVA independent variables. multiple factors. Mann-Whitney U Test Check if group variances are Tests assumption of equal Levene's Test equal before ANOVA/t-test. variances. Fisher's Score Test if group means are Ratio of explained to Mann-Whitney U Test: Used to compare two independent groups. It's (F-test) different (in ANOVA). unexplained variance. often used when the data doesn't meet the assumptions of parametric tests Compare means of 2 groups Use if Levene's test shows like the t-test. Welch's t-Test with unequal variances. unequal variances. Test if variances between groups are When to Use? Sphericity Requires correction if violated. equal (Repeated Measures ANOVA). Non-normal Data: Data is not normally distributed. Greenhouse-Geisser Adjust for sphericity violations Ordinal Data: Data is measured on an ordinal scale. Lowers degrees of freedom. Correction (conservative). ε < 0.75 Small Sample Sizes: Sample sizes are small (less than 30). Huynh-Feldt Adjust for sphericity violations Use for moderate violations. Correction (aggressive). ε > 0.75 Assumptions: Find out which groups differ after Used for pairwise Independence: Observations within each group are independent. Tukey Post-Hoc significant ANOVA. comparisons. Random Sampling: Samples are drawn randomly from the respective Measure the relationship between Pearson’s Correlation Linear, normal data. populations. 2 continuous variables. Simple Linear Predict one variable based on Assumes linearity and Interpretation: Regression another. normality. If the calculated U value is statistically significant, it suggests that Multiple Linear Predict one variable using 2+ Same as simple, but with more there is a difference between the two groups. However, it does not indicate Regression predictors. variables. the nature or magnitude of the difference. 14.0 Non-parametric Tests Wilcoxon Signed-Rank Test Non-parametric Tests: Statistical tests that do not make strong assumptions about the underlying distribution of the data. They are useful Wilcoxon Signed-Rank Test: Used to compare two related samples. when the data does not meet the assumptions required for parametric When to Use? tests, such as normality or homogeneity of variance. Non-normal Data: Data is not normally distributed. No Normality Assumption: Does not require data to follow a normal Ordinal Data: Data is measured on an ordinal scale. distribution. Paired Samples: When you have two measurements from the same Flexible Data Types: Can be used with ordinal, nominal, interval, or individuals or paired units. ratio data. Interpretation: 14.1 When to Use Non-parametric Tests If the test statistic is less than or equal to the critical value, you reject the null hypothesis and conclude that there is a significant difference between Unknown Distribution: When the population distribution is the two related samples. unknown or cannot be assumed. Ordinal or Nominal Data: When the data is not continuous or Kruskal-Wallis Test normally distributed Outliers: When there are extreme values that can skew the results of Kruskal-Wallis Test: Used to compare three or more independent groups parametric tests. to determine if there are statistically significant differences between them. Small Sample Size: When the sample size is too small (less than 30) It's a rank-based test, meaning it doesn't rely on the assumption of to assume normality. normality in the data. Akin towards a non-parametric ANOVA. 14.2 Advantages and Disadvantages Interpretation: Small p-value: If the p-value is less than the significance level (0.05), it Advantages: indicates that there is a significant difference between at least two of the groups. Fewer Assumptions: No strict requirements for normality or equal variances. Large p-value: If the p-value is greater than the significance level (0.05), Robust to Outliers: Less sensitive to outliers, providing more reliable it indicates that there is no significant difference between the groups. results in skewed distributions. Versatile: Can be applied to various data types. she 🐧 | Review Well~ Course Enrichment I Page 9 Post-hoc Tests: Types of Chi-Square Tests: If the Kruskal-Wallis test is significant, post-hoc tests can be used to Chi-Square Test of Independence: Used to determine if there is a identify which specific groups differ from each other. relationship between two categorical variables. Common post-hoc tests: For example, you could use it to test if there is a relationship between Dunn’s multiple comparisons test gender and preference for a particular brand of car. Conover-Inman test Chi-Square Test of Goodness of Fit: used to determine if the observed distribution of a single categorical variable matches an expected distribution. Friedman Test For example, you could use it to test if the number of people who prefer different colors of cars follows a uniform distribution. Friedman Test: Used to determine if there are differences in the medians of three or more related groups. Summary When to Use? Ordinal or Continuous Data: Data is either ordinal or continuous. Test When to Use? Key Points Related Samples: The same individuals or groups are measured multiple Use for ordinal or Compare two independent groups times. Mann-Whitney U Test non-normally distributed data; (non-parametric version of t-test). no need for equal variances. Violated Assumptions of ANOVA: If normality or homogeneity of Compare two related groups Use for paired data (e.g., variances are not met. Wilcoxon Signed-Rank (non-parametric version of paired before/after) when data is not Test t-test). normal. Advantages: Compare three or more Robust to Assumptions: Doesn’t require normality or homogeneity of Use for ordinal or non-normal independent groups variances. Kruskal-Wallis Test (non-parametric version of data; tests for differences between groups. Easy to Apply: Relatively straightforward to calculate and interpret. ANOVA). Handles Ordinal Data: Can be used with ranked data. Compare three or more related Use for repeated measures Friedman Test groups (non-parametric version of with ordinal or non-normal Disadvantages: repeated measures ANOVA). data. Less Powerful: The Friedman test is less effective than ANOVA in Test for association between Compares frequencies in Chi-Square Test different categories to see if detecting differences when ANOVA's assumptions are met. categorical variables. they differ significantly. Limited Follow-up Tests: After a significant result, the Friedman test has fewer follow-up options compared to ANOVA. Parametric vs Non-parametric Friedman Test vs Kruskal-Wallis Test Question If yes, then use If no, then use Key Differences: Is your data normally distributed? Parametric Test Non-parametric Test Data Structure: Kruskal-Wallis is for independent groups, while Non-parametric Test Is your data interval/ratio? Parametric Test Friedman is for repeated measures or matched groups. (if ordinal/nominal) Within-Subjects vs. Between-Subjects: Friedman is a within-subjects Are variances equal across groups? Parametric Test Non-parametric Test test, while Kruskal-Wallis is a between-subjects test. (homoscedasticity) (T-test/ANOVA). (Welch’s test) Independent t-test / Paired Test (Paired t-test, Statistical Test: Kruskal-Wallis is a generalization of the Mann-Whitney Are your samples independent? ANOVA Wilcoxon Signed-Rank) U test, while Friedman is a generalization of the sign test. Do you have outliers or skewed Parametric Test Non-parametric Test data? (violates assumptions) (if normally distributed) Example: Is your sample size large enough Parametric Test (if Non-Parametric Test Kruskal-Wallis: Comparing the median incomes of three different (>30)? normality holds) (better for small samples) professions (e.g., doctors, lawyers, teachers) Do you need to compare two Mann-Whitney U Friedman: Comparing the performance of a group of students on a test groups? t-test (parametric) (non-parametric) administered at three different time points (e.g., before, during, and after Do you need to compare three or One-Way ANOVA Kruskal-Wallis Test a course). more groups? (parametric) (non-parametric) Are you working with repeated Repeated Measures Friedman Test measures? ANOVA (non-parametric) Chi-Square Test Chi-Square Test: Used to determine and compare how closely your observed data matches the expected distribution. When to Use? Categorical Data: Data falls into distinct categories or groups (yes/no). Comparing Observed Frequencies to Expected Frequencies: For instance, you might want to compare the number of people who prefer different brands of soda to the expected distribution based on market share. she 🐧 | Review Well~ Course Enrichment I Page 10