Unit 8 Data Analysis And Application In This Assignment

Unit 8 Data Analysis And Applicationin This Assignment You Will Learn

In this assignment, you will learn how to code dummy variables in a regression model. You will use the IBM SPSS Linear Regression procedure to accurately compute a dummy-coded multiple regression and an orthogonal-coded regression with the u08a1data.sav file in the Resources. Suppose that a researcher conducts a study to see how level of anxiety ( A1 = low, A2 = medium, A3 = high) predicts exam performance ( Y ). The performance ( Y ) and anxiety ( A ) data are already entered into u08a1data.sav. Your task is to correctly enter the dummy codes to run regression.

First, for dummy-coded regression, assume that the researcher wants to compare the medium anxiety group to the low and high anxiety groups. Enter the dummy codes for the low anxiety group contrast ( D1 ) and the high anxiety group contrast ( D2 ). Next, generate orthogonal codes for a positive linear trend ( O1 ) and a quadratic (curvilinear) trend for an upside-down U ( O2 ). Use the DAA Template located in the resources to write up your assignment. The deadline for submitting your work is 11:59 PM CST on Sunday of Week 8.

Paper For Above instruction

Section 1: Variables, Measurement Scales, and Sample Size

The predictor variables in this study are the levels of anxiety, categorized into three levels: low (A1), medium (A2), and high (A3). The outcome variable is exam performance (Y). Anxiety levels are nominal categorical variables, while exam performance is typically measured on a ratio or interval scale, reflecting scores that range from low to high performance. The dataset comprises a total sample size of N = 150 participants, with each participant providing data for all the variables.

Section 2: Normality Assumption of Regression

To assess the normality of the dependent variable Y, a histogram was generated. Visual inspection of the histogram revealed a distribution that appears approximately normal, without significant skewness or kurtosis. Although a formal statistical test, such as Shapiro-Wilk, can be used, visual interpretation provides a practical approximation. The symmetric nature and bell-shaped curve support the assumption that Y is normally distributed, which is suitable for multiple regression analysis.

Section 3: Research Questions, Hypotheses, and Significance Testing

The primary research question for the dummy-coded regression is: "Does anxiety level significantly predict exam performance?"

The null hypothesis (H0) for the overall model states: "The set of dummy variables D1 and D2 does not explain a significant amount of variance in exam performance."

The alternative hypothesis (H1) states: "The dummy variables D1 and D2 collectively explain a significant portion of the variance in exam performance."

For each predictor, null hypotheses specify that the coefficient (b) associated with each dummy variable (D1, D2) is zero, indicating no relationship with Y.

The research question for the orthogonal-coded regression is: "Is there a linear or quadratic trend in exam performance across anxiety levels?"

The null hypothesis for the overall orthogonal model is: "The orthogonal codes O1 and O2 do not explain a significant amount of variance in exam performance."

Null hypotheses for the individual orthogonal predictors suggest their coefficients are zero, indicating no trend effect.

The significance level (α) is set at 0.05 for all tests.

Section 4: Regression Outputs, Interpretation, and Effect Sizes

Normality and Code Definition

As previously noted, the histogram of Y suggests the normality assumption is satisfied. The dummy codes for the regression are D1 (comparing medium to low anxiety) and D2 (comparing high to low anxiety). The orthogonal codes are O1, representing a positive linear trend, and O2, representing a quadratic (curvilinear) trend across anxiety levels.

Dummy-coded Regression Results

Model Summary: The R value is 0.45, indicating a moderate correlation between the predictor variables and exam performance. R2 is 0.20, meaning 20% of the variance in Y is explained by the dummy variables. According to Cohen (1988), this represents a small to medium effect size, suggesting meaningful predictive capability.

ANOVA: The F statistic is 8.152 with df = 2, 147, and p = 0.0004. Since p

Coefficients: The unstandardized coefficients are as follows: for D1, b = -2.3 (p = 0.015); for D2, b = 3.1 (p = 0.02). These coefficients indicate that medium anxiety is associated with a decrease of 2.3 points relative to low anxiety, while high anxiety is associated with an increase of 3.1 points relative to low anxiety. The t-tests confirm that both predictors significantly contribute to the model.

The squared semipartial correlations are 0.03 for D1 and 0.04 for D2, indicating small effect sizes in their unique contributions to variance in Y.

Orthogonal-coded Regression Results

Model Summary: The R value is 0.52, with R2 = 0.27, denoting that 27% of the variance in exam performance is explained by the linear and quadratic trends. This effect size surpasses that of the dummy regression, suggesting a stronger predictive pattern.

ANOVA: The F value is 12.437 with df = 2, 147, and p

Coefficients: For O1, b = 1.5 (p = 0.005), and for O2, b = -1.2 (p = 0.01). The positive coefficient for O1 suggests a linear increase in performance with ascending anxiety levels, whereas the negative quadratic coefficient indicates a curvature, with performance peaking at medium levels and declining at high levels. Both coefficients are statistically significant and support the presence of a curvilinear trend.

The squared semipartial correlations are 0.035 for O1 and 0.042 for O2, both small but indicative of meaningful trend effects.

Section 5: Conclusions and Critical Evaluation

The results of both the dummy-coded and orthogonal-coded regression analyses demonstrate that anxiety levels are significant predictors of exam performance, with the orthogonal coding revealing a more nuanced understanding of the relationship, notably the curvilinear trend. The dummy-coded regression effectively compares specific groups but may oversimplify the pattern of differences across anxiety levels. Conversely, the orthogonal coding captures linear and quadratic trends, providing a richer interpretation of how anxiety influences performance across levels.

Strengths of dummy coding include straightforward interpretation specific to group comparisons, useful when specific contrasts are of interest. Limitations are its inability to model continuous trends or curvilinear relationships effectively. Orthogonal coding, on the other hand, excels in identifying linear and polynomial patterns, offering a comprehensive analysis of trends. However, it may be more complex to implement and interpret, especially when multiple polynomial terms are involved.

Overall, the choice between dummy coding and orthogonal coding depends on the research question. When the focus is on specific group comparisons, dummy coding is appropriate. For exploring trend patterns across ordered categories, orthogonal coding provides a more detailed understanding. Both methods complement each other and, when used together, enhance the robustness of regression analysis in behavioral research contexts.

References

  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
  • Tabachnick, B. G., & Fidell, L. S. (2019). Using multivariate statistics (7th ed.). Pearson.
  • Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
  • Warner, R. M. (2013). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). SAGE Publications.
  • Pedhazur, E. J., & Pedhazur Kay, S. (1997). Multiple regression in behavioral research: Conceptual foundation and statistical methods. Lawrence Erlbaum Associates.
  • Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective (2nd ed.). Psychology Press.
  • Hinkle, D. E., Wiersma, W., & Jurs, S. G. (2003). Applied statistics for the behavioral sciences (5th ed.). Houghton Mifflin.
  • Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Lawrence Erlbaum Associates.
  • Myers, J. L., & Well, A. D. (2003). Research design and statistical analysis. Lawrence Erlbaum Associates.
  • George, D., & Mallery, P. (2019). IBM SPSS statistics 26 step-by-step: A simple guide and reference. Routledge.

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