Data Analysis & Statistics for Social Work — Study Notes

Motivating Scenarios

• Sheldon (Descriptive Study)

  • Task: prepare client-profile section for a grant, aiming to understand the current client base.

  • Data come from current records for $n = 234$ clients; one data sheet per client.

  • Six characteristics recorded:

    • Proportion male vs. female (binary demographic).

    • Proportion in each race category (categorical demographic).

    • Mean age (continuous numerical data).

    • Mean income (continuous numerical data).

    • Number served in each service category (count data, reflecting utilization).

    • Number of families with $\ge 1$ preschool child (binary or count, indicating family structure needs).

  • Uses statistical-analysis software to compute descriptive statistics (frequencies, proportions, means), which summarize and present the features of the client base without making inferences beyond the sample.

• Dana (Explanatory Study)

  • Question: Which variables explain client no-shows? This involves identifying predictive factors.

  • Five dichotomous predictors (i.e., variables with two categories, like yes/no):

    • 3 self-identified (e.g., client reports being employed/unemployed, having transportation/not).

    • 2 from literature (e.g., history of mental health diagnosis/no diagnosis, distance from clinic: near/far).

  • Samples: repeat no-shows vs. perfect attenders. This creates two distinct groups for comparison.

  • Statistic: Chi-square ($\chi^2$) test (or Fisher exact for small samples) because all variables are dichotomies and the goal is to examine associations between categorical variables.

  • Interpretation rule: if the no-show proportion is "significantly" higher on a variable (p < .05), treat as a causal factor. This implies a statistical association, not necessarily direct causality without a robust research design.

• Hanna (Evaluative Single-Case Study)

  • Client: newly divorced man with depression. This is an N=1 study focused on individual change.

  • Measures: standardized depression scale at baseline (before intervention) + before each of six weekly sessions (during intervention). This allows for monitoring progress over time.

  • Statistic: one-sample $t$ test (compare 6 treatment scores vs. 1 baseline score), which evaluates if the treatment scores significantly differ from the pre-treatment baseline.

Three Core Uses of Data Analysis in Social Work

• Describe: Focuses on summarizing the characteristics of a sample or population. (Descriptive stats: means, frequencies, proportions, medians, modes, ranges, SDs). Provides a snapshot of the data.

• Explain: Aims to understand relationships between variables, often involving cause-and-effect hypotheses. (Explanatory research with inferential stats: correlations, $t$-tests, $\chi^2$-tests, ANOVA, regression, etc.). Seeks to generalize findings from a sample to a larger population.

• Evaluate: Assesses the effectiveness of interventions, programs, or policies. (Practice/outcome evaluation with inferential stats: paired/independent/one-sample $t$-tests, effect size, SD method, etc.). Determines if changes occurred and how meaningful they are.

Review of Prior Concepts

• Research phases: a systematic process including:

  • question: formulating a clear, researchable inquiry.

  • knowledge/methods: reviewing existing literature and choosing appropriate research designs and statistical approaches.

  • data collection: systematically gathering relevant information.

  • data analysis: processing and interpreting collected data using statistical methods.

  • conclusions: drawing informed inferences based on the analysis and addressing the initial research question.

• Statistical vs. practical significance:

  • Statistical: probability results are due to chance ($p$-value). A low $p$-value (e.g., p < .05) suggests the observed effect is unlikely to be random. It doesn't indicate the magnitude or importance of the effect.

  • Practical/clinical: whether change is meaningful or important in real life. Even a statistically significant finding might have little practical relevance if the effect size is very small.

• Levels of measurement: dictate which statistical analyses are appropriate.

  • Nominal: Categories with no inherent order (e.g., gender, race).

  • Ordinal: Categories with a meaningful order but unequal intervals between them (e.g., Likert scales: strongly agree, agree, neutral).

  • Interval (ratio treated as interval in text): Ordered data with equal intervals between values but no true zero point (e.g., temperature in Celsius). Ratio variables have a true zero (e.g., income, age) allowing for ratio comparisons; for most statistical purposes, they are handled similarly to interval data.

• Hypothesis testing requires: a stated direction (e.g., Group A will score higher than Group B) + statistical significance (a $p$-value meeting a predetermined threshold, usually $.05$).

• Conclusions must mirror data; avoid opinion masquerading as evidence or overstating the implications of findings beyond what the data supports.

Descriptive Statistics Toolkit

Nominal-Level Variables

• Frequency ($n$) = raw count of observations within each category. Useful for understanding distribution within discrete categories.

• Proportion/percentage: $\% = \frac{f}{N} \times 100$ (where $f$ is frequency of a category and $N$ is total observations). Provides a standardized way to compare parts to the whole.

• Visualization: Bar charts are ideal for displaying frequencies or proportions of categorical data, showing the relative size of each category.

Ordinal-Level Variables

• All nominal stats plus Median (middle value when data is ordered low$\rightarrow$high). The median is preferred for ordinal data because it is not affected by extreme values and represents the central tendency based on ranked position.

• Example: Ages ${15,17,19,19,23,28,33,44,55}$ (sorted) $\rightarrow$ median $= 23$. If there's an even number of values, the median is the average of the two middle values.

Interval/Ratio-Level Variables

• All lower-level stats plus:

  • Mode (most common value). Useful for identifying the most frequent observation, but can be unstable if data distribution is flat.

  • Mean: $\bar{x} = \frac{\sum x_i}{N}$; (sum of all values divided by the number of values). The most common measure of central tendency for interval/ratio data, but sensitive to outliers.

  • Range: $\text{Max} - \text{Min}$ (example $55-15=40$ years). Provides a simple measure of spread, but only uses two data points and is highly sensitive to outliers.

  • Standard Deviation (SD): Measures the average dispersion of data points around the mean. A small SD indicates data points are clustered closely around the mean, while a large SD indicates greater variability.

Inferential Statistics for Explanatory Research

Key Selection Questions

1. Number of variables: Determines if you are looking at univariate (one variable), bivariate (two variables), or multivariate (three or more variables) relationships.

2. Measurement level of each variable: Crucial for selecting the appropriate statistical test, as different tests are designed for different data types (nominal, ordinal, interval/ratio).

3. Identify dependent vs. independent variables (for $\ge 3$ variables): In explanatory research, distinguishing which variable is hypothesized to cause or influence another (independent variable) from the variable that changes in response (dependent variable) is fundamental.

4. Independence of samples (paired vs. independent): Determines if observations in one group are related to observations in another (e.g., pre-test/post-test are paired) or if groups are entirely separate (e.g., treatment group vs. control group are independent).

Common Data Structures & Statistics (two-variable hypotheses)

1. Two dichotomies (e.g., Gender and Employment Status) $\rightarrow \chi^2$ (large $N$) / Fisher exact (small $N$). Used to test for association between two categorical variables.

2. Two nominal variables (any categories, e.g., Religion and Preferred Therapy Type) $\rightarrow \chi^2$. Extends the test of association to categorical variables with more than two categories.

3. Dichotomy (Independent Variable, IV) + interval score (Dependent Variable, DV) (e.g., Male/Female and Depression Score) $\rightarrow$ Independent $t$ test. Compares the means of a continuous variable between two independent groups.

4. Nominal (>2 categories, IV) + interval (DV) (e.g., Marital Status (single, married, divorced) and Anxiety Score) $\rightarrow$ ANOVA (Analysis of Variance). Compares the means of a continuous variable across three or more independent groups.

5. Two interval scores (e.g., Income and Life Satisfaction Score) $\rightarrow$ Pearson $r$ (correlation coefficient). Measures the strength and direction of the linear relationship between two continuous variables.

6. Two ordinal variables (e.g., Education Level (high school, college, grad) and Client Satisfaction Rank) $\rightarrow$ Spearman $\rho$ (correlation coefficient). Measures the strength and direction of the monotonic relationship between two ordinal variables.

7. Interval DV + $\ge 2$ interval IVs (e.g., Explaining Life Satisfaction with Income, Education, and Age) $\rightarrow$ Multiple regression. Predicts the value of a dependent variable based on the values of two or more independent variables.

Interpreting Outputs

• Tables of two nominal vars: compare row/column proportions to see patterns, then check $p$ via $\chi^2$ test to determine if the observed association is statistically significant. For example, comparing the percentage of males in one outcome category vs. females.

• Correlation: Pearson $r$ ranges from $-1$ to $+1$.

  • The absolute value ($0 \rightarrow 1$) indicates the strength of the linear relationship (closer to 1 is stronger).

  • The sign (positive vs. negative) indicates the direction of the relationship.

    • Positive relationship: as one variable increases, the other tends to increase (e.g., higher social support $\rightarrow$ higher life satisfaction).

    • Negative relationship: as one variable increases, the other tends to decrease (e.g., higher depression $\rightarrow$ lower academic grades).

  • Perfect positive example: grades Exam1 vs. Exam2 $r = 1.0$ (a perfectly linear relationship where as one score increases, the other increases proportionally; the line slopes up).

• $t$ value: A test statistic used to locate $p$-value by comparing the means of two groups relative to the variability within the groups. It is not a direct strength indicator of the relationship itself.

• Effect size (Cohen’s $d$): A standardized measure of the magnitude of an observed effect, expressed in standard deviation units. This allows for comparing the practical significance of findings across different studies or measures. Common interpretations: $d=0.2$ (small), $d=0.5$ (medium), $d=0.8$ (large).

Inferential Statistics for Evaluative Research

Exhibit-Based Quick Guide

1. One-group pretest–posttest (matched scores) $\rightarrow$ Paired $t$ test. Used when the same group of individuals is measured before and after an intervention (e.g., depression scores before and after therapy for the same clients).

2. One-group pretest–posttest (unmatched; use mean pretest) $\rightarrow$ One-sample $t$ test. Used when individual baseline scores are unavailable, but an overall mean baseline score is known, and individual post-test scores are compared to this known mean.

3. Comparison-group design (gain scores) $\rightarrow$ Independent $t$ test. Used to compare the average change (gain) in scores between two independent groups (e.g., a treatment group vs. a control group).

4. Limited AB single-case (one baseline score vs. multiple treatment scores) $\rightarrow$ One-sample $t$ test. Similar to the scenario above, useful when a single baseline measurement is compared against a series of measurements taken during an intervention phase.

5. AB single-case (multiple baseline & treatment scores) $\rightarrow$ SD method. This method compares treatment phase data to the variability of the baseline phase data to determine if a meaningful change has occurred.

   • Compute baseline Standard Deviation ($\text{SD}<em>{base}$) and mean ($\bar{x}</em>{base}$) from multiple baseline data points.

   • Criterion = $\bar{x}<em>{base} + 2 \times \text{SD}</em>{base}$ (for improvement in the positive direction). This sets a threshold beyond which a score is considered significantly different from the baseline variability.

   • If $\bar{x}_{treat}$ (mean of treatment scores) surpasses this calculated criterion (for the improvement direction), then the change is considered statistically significant, approximately equivalent to p < .05. This provides a clear rule for determining treatment efficacy in single-case designs.

Reporting Template Examples

• Provide practical indicators (e.g., means, counts, proportions, correlations) that describe the observed patterns in the data, alongside statistical indicators ($n$, test statistics like $t, \chi^2, r$, and the $p$-value) that justify the statistical significance.

• Example paired $t$ write-up: “There was a statistically significant drop in anxiety scores from pre-test to post-test. Anxiety fell from a mean of $\bar{x}<em>{pre}=24.5$ ($\text{SD}=5.2$) to a mean of $\bar{x}</em>{post}=17.1$ ($\text{SD}=4.8$); t(16)=2.21, p<.05. Client anxiety levels decreased significantly after the intervention.”

Guidelines for Writing Conclusions

• Begin with a succinct restatement of the research question and the key data pattern observed. This provides immediate context for the reader.

• Do not over-generalize findings beyond the sample studied or introduce unsupported opinions. Conclusions should be strictly data-driven.

• If p > .05, clearly state that the hypothesis was not supported, even if directional trends were observed. Avoid implying significance where none exists statistically.

• Acknowledge limitations: Discuss potential biases in sampling (e.g., non-random), measurement choices (e.g., reliability/validity issues of scales), and design threats to validity (e.g., internal validity threats like history or maturation, or external validity threats impacting generalizability).

• Discuss practice implications (e.g., for intervention development, policy changes): This is where findings are translated into actionable insights for social work practice (e.g., suggesting reminder calls improve attendance, offering night hours increases accessibility, or focusing treatment on specific client needs).

• Maintain scientific caution; single studies contribute incrementally to the broader evidence base, and their findings should be viewed as part of ongoing scientific inquiry, not definitive truths.

Practical/Technical Tips

• Case-level data (one row per client, all variables organized in columns) is required before analysis. This structured format (e.g., spreadsheet) is essential for most statistical software.

• Descriptive studies analyze one variable at a time (e.g., mean age of clients, proportion of males); explanatory/evaluative studies look at relationships or comparisons between two or more variables.

• Dichotomous variables (two categories, e.g., yes/no, male/female) sometimes receive special handling in tests (e.g., they can be used as independent variables in $t$-tests or as dummy variables in regression).

• Internet statistical tools or basic spreadsheet functions can suffice for simple projects, especially for descriptive statistics and very basic inferential tests.

• Selecting the wrong level of measurement (e.g., treating ordinal data as interval data) is a minor error in everyday practice for some exploratory analyses but should be avoided when possible, especially in formal research, as it can lead to inaccurate statistical conclusions or misinterpretations.