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Textbook, 2013, 131 Pages
Although statistics is not always a popular subject, perhaps because it relies heavily upon mathematics, statistical skills are necessary to anyone intending to practice medicine or engage in health-related research or in medical care service delivery. Covering every step of some of the most commonly used data analysis techniques, this book offers a road map and strategies to adopt when medical students, doctors, or other health professionals face challenges in handling and analyzing clinical or survey data - from preliminary evaluations of the databases in question to how to effectively interpret the results.
The book is divided into two sections, each is packed with distinct perspective on the database assessed, taking the reader through the process of developing answerable and relevant research questions and their corresponding hypotheses. Using SPSS, chapters 1-5 analyze data derived from a noise-testing scores database by presenting descriptive statistics and statistical hypothesis testing, including potential templates for presenting overviews of the study designs, guidance on how to choose appropriate tests, ways to understand their assumptions, guidelines on how to correctly check them, and which limitations are important to consider. In chapters 6-9, another set of statistical analyses are carried out utilizing data derived from the Ugandan database, which provides records of the perception of chances of getting AIDS among Ugandan women. The effects of region, place of residence, and condom-use on ages at first marriage and intercourse are investigated thoroughly. General conclusions and observations are recorded in the final chapter.
Addressing most, if not all, aspects of ANOVAs, ANCOVAs, MANOVAs, MANCOVAs, Chi-square tests, and t -tests, this book is designed as a toolkit with much-needed directions to aid those who have to enter the world of medical statistics.
Key words: Medical research, ANOVA, ANCOVA, MANOVA, MANCOVA, Chi-square, t-test, database, assumption checking, research questions, hypotheses, SPSS, biostatistics, public health
Please cite this publication as (APA style):
Akrawi. W. (2013). Statistical evaluation of two medical databases: A framework on descriptive, bivariate, and multivariate analyses. Germany: GRIN Publishing.
Intended Audience
- Professional and scholarly (e.g., doctors with interest in statistics, educators, researchers);
- College or higher education (e.g., master's and PhD students); and
- General (e.g., anyone interested in hands-on experience on data analysis techniques using SPSS).
BOOK DESCRIPTION
CITATION AND INTENDED AUDIENCE
Citation
Intended Audience
LIST OF TABLES
LIST OF FIGURES
LIST OF ABBREVIATIONS
INTRODUCTION
Book Highlights
Main Objectives
PART ONE: THE NOISE-TESTING SCORES DATABASE
CHAPTER 1: OVERVIEW OF DATABASE
Study Design
Descriptive Statistics
CHAPTER 2: BIVARIATE ANALYSES
T-Tests
Research question 1
Research question 2
Joint research question
Results
Chi-Square Tests
Research question 1
Research question 2
Results
Interpretation
CHAPTER 3: ANALYSES OF VARIANCE
ANOVA Assumptions
Effects of Parents' Highest Level of Education on Test Scores
Research questions and corresponding hypotheses
Results
Effects of Gender and Type of Noise on Reading Test Scores
Research question and corresponding hypotheses
Results
Effects of Gender and Type of Noise on Math Test Scores
Research question and corresponding hypotheses
Results
CHAPTER 4: MULTIVARIATE ANALYSIS
Hypothesis Raised
Results
CHAPTER 5: DISCUSSION ON NOISE DATABASE
PART TWO: THE UGANDAN DATABASE
CHAPTER 6: OVERVIEW OF DATABASE
Relevance of Study
Study Design
Study Sample
Descriptive Data Analysis
CHAPTER 7: BIVARIATE ANALYSIS
T-Tests
Research questions and corresponding hypotheses
Results
Chi-Square Tests
Research question and corresponding hypotheses
Results
Interpretation
Summary
One-Way and Two-Way ANOVAs
ANOVA
Summary
Effects of perception of chances of getting AIDS on age at first marriage
Effects of Region on Age at First Marriage
Effects of place of residence on age at first marriage
Effects of region and residence on age at 1st marriage
Effects of condom use and perception of aids risk on age at first marriage
Effects of residence and perception of aids risk on age at first marriage
Effects of region and ever use of condoms on age at first marriage
ANOVAs and age at first sexual intercourse
Effects of region and perception of aids risk on age at first intercourse
Effects of ever use of condoms and region on age at first intercourse
Effects of region and ever use of condoms on age at first marriage with a covariate
Effects of region on age of subject
CHAPTER 8: MULTIVARIATE ANALYSES
Effects of Region and Condom Use on Ages at First Marriage and Intercourse
Hypotheses raised
Results
Effects of Region, Condom-Use and Perception of Chances of Getting AIDS on Ages at First
Marriage and Intercourse
Hypotheses raised
Results
CHAPTER 9: DISCUSSION ON UGANDAN DATABASE
CHAPTER 10: CONCLUSIONS
REFERENCES
INDEX
Table 1. Crosstabulation of Parents' Highest Level of Education by Noise Type
Table 2. Descriptive Statistics for Reading and Math Test Scores
Table 3. Post Hoc Tests (Bonferroni) for the Effects of Parent' Highest Level of Education Completed on Students' Reading Test Scores
Table 4. Post Hoc Tests (Bonferroni) for the Effects of Parent' Highest Level of Education Completed on Students' Math Test Scores
Table 5. ANOVA for the Effects of Noise Type and Gender on the Students' Reading Scores .. 25 Table 6. Post Hoc Tests (Bonferroni) for the Effects of Gender and Noise Type on Students' Reading Test Scores
Table 7. ANOVA for the effects of Noise Type and Gender on the Students' Math Scores
Table 8. Post Hoc Tests (Bonferroni) for the Effects of Gender and Noise Type on Students' Math Test Scores
Table 9. Correlation between Reading Test Scores and Math Test Scores
Table 10. Multivariate Test for Reading Test Scores and Math Test Scores by Noise Type and Gender
Table 11. ANOVA for the Effects of Type of Noise and Gender on Reading Test Scores and Math Test Scores
Table 12. Crosstabulation of Ever Use of Condoms by Residence in the Ugandan Population ..
Table 13. Descriptive Statistics for Selected Variables in the Ugandan Database
Table 14. Post Hoc Tests (Bonferroni) for the Effects of Perception of Chances of Getting AIDS on Age at First Marriage among Ugandan Women
Table 15. ANOVA for the Effects of Residence and Region on Age at First Marriage
Table 16. ANOVA for the Effects of Perception of the Chances of Getting AIDS and Condom Use on Age at First Marriage
Table 17. Post Hoc Tests (Bonferroni) for the Effects of Condom Use on Age at First Marriage among Ugandan Women
Table 18. ANOVA for the Effects of Perception of the Chances of Getting AIDS and Residence on Age at First Marriage
Table 19. Two-Way ANOVA for Age at First Marriage by region and Condom Use
Table 20. ANOVA for the Effects of Region and Perception of the Chances of Getting AIDS on Age at First Sexual Intercourse
Table 21. Post Hoc Tests (Bonferroni) for the Effects of Region on Age at First Sexual Intercourse among Ugandan Women
Table 22. Two-Way ANOVA for Age at First Sexual Intercourse by Region and Condom Use 75 Table 23. ANCOVA for Age at First Marriage by Region and Condom Use, with Age at First Sexual Intercourse as a Covariate
Table 24. Post Hoc Tests (Bonferroni) for the Effects of Region on Age of Subject among Ugandan Women
Table 25. Correlation between Age at First Marriage, Age at First Sexual Intercourse and Age of Subject
Table 26. Multivariate Test for Age at First Marriage and Age at First Sexual Intercourse by Region and Condom Use
Table 27. ANOVA for the Effects of Region and Condom Use on Age at First Marriage and Age at First Sexual Intercourse
Table 28. Multivariate Test for Ages at First Marriage and Sexual Intercourse by Region, Condom Use and Perception of the Chances of Getting AIDS
Table 29. ANOVA for the Effects of Region, Condom Use and Perception of the Chances of Getting AIDS on Age at First Marriage and Age at First Sexual Intercourse
Figure 1. Estimated marginal means of reading test scores for students tested by noise type and gender
Figure 2. Estimated marginal means of math test scores for students tested by noise type and gender
Figure 3. Estimated marginal means of age at first marriage for the Ugandan women by residence and region
Figure 4. Estimated marginal means of age at first marriage for the Ugandan women by region
and condom Use
Figure 5. Estimated marginal means of age at first sexual intercourse for the Ugandan women by region and perception of the chances of getting AIDS
Figure 6. Estimated marginal means of age at first sexual intercourse for the Ugandan women by region and ever condom use
illustration not visible in this excerpt
The statistical analyses used in this book are some of the most frequently applied analyses in (medical) research. For instance, Analysis of Variance (ANOVA) is used extensively in many areas of research, such as medicine, psychology, biology, education, sociology, and political science (Rutherford, 2001). One of the reasons behind this is its suitability for many different types of study design, including experimental and quasi-experimental designs (Sim & Wright, 2000; Rutherford, 2001). Additionally, ANOVA does not impose any restriction on the number of groups or conditions that may be compared, and factorial ANOVA allows examination of the effects of two or more independent variables or factors on the dependent variable (Rutherford, 2001). Factorial designs, in particular, also have other advantages. Using this kind of study to test several hypotheses in a single research study has impacts on the feasibility of the study because it is more cost effective to conduct a single factorial design than conducting several individual studies (Marczyk et al., 2005). Besides, factorial studies offer a great flexibility to explore or enhance the "signal" of interest (Trochim, 2006). Also, factorial designs provide a framework that allows examination of the interactions between the independent variables (Marczyk et al., 2005; Trochim, 2006).
Two other complex versions of ANOVA and ANCOVA are Multivariate Analysis of Variance (MANOVA) and a MANOVA with a covariate (MANCOVA), respectively. These tests are, computationally, more complex and time-consuming than ANOVA and ANCOVA. In simple terms, MANOVA is a multiple ANOVA, and it is performed to examine two or more continuous dependent variables in relation to the groups formed by categorical independent variables (Cramer & Howitt, 2004). It is employed instead of multiple ANOVAs to prevent an inflation of the familywise or experimentwise error rates, i.e., to reduce the chance of making one or more Type I errors caused by running multiple tests on the same data (Field, 2005).
MANOVA tests the hypothesis that two or more (continuous) dependent variables are affected by the difference in independent variables (StatSoft, 2010). Furthermore, MANCOVA is similar to MANOVA, but a "covariate" is included in the model as it may correlate with the response variables and this correlation must be taken into account when performing the significance test (StatSoft, 2010).
Being able to identify the research questions is vital for the process of choosing the appropriate statistical test and this, in turn, is vital to being able to extract maximum information from the data at hand. Research questions are derived from the broad aims of the research study; hence they differ from hypotheses in that they are not derived as specific predictions from theories or statistical inference procedures (Hall, 2008). It should also be noted that failure to specify the relevant research questions and hypotheses may jeopardize the study and lead to misleading results.
In this book, statistical tests are performed on two real-world databases. The book provides a practical framework that can be followed in reporting the descriptive statistics, checking the assumptions, performing bivariate and multivariate tests, reporting the results, interpreting and discussing them, and presenting the conclusions drawn. It introduces key concepts (e.g., how to master the field of medical statistics and quantitative research); presents simple, clear procedures, with ready-to-use formulated null and alternative hypotheses; and offers a simple toolkit, with numerous statistical resources, needed to analyze several continuous dependent variables with respect to various independent variables. This book can be of interest for medical students, doctors, PhD students, health professionals, educators, medical researchers, and others who might be interested in knowledge about the way in which statistics and clinical practice (e.g., disease control or prevention) are interconnected.
As implied above, this book is divided into two parts. The first part involves performing various statistical tests on data collected using a 3 x 2 factorial study, with three type of noise (classical, traffic, or quiet) and two performance tests (math or reading) to determine if students performed better on examination in math and reading when they were exposed to classical music (Chopin or Mozart). The database that was used to test the hypotheses proposed in this part was named Noise-Testing Scores. The results highlighted that students hearing classical music or learning in quiet environments performed significantly better than those disrupted by traffic noise.
The second part of the book explores the population sampled for the Ugandan database named Ugandan-SPSS-Data.sav. Analyzing data from a survey of 1479 Ugandan women on their perception of the chances of getting HIV/AIDS, this part uses descriptive, bivariate and multivariate analyses to assess the relationships between variables stored in the database. The relevance of the Ugandan survey is emphasized in chapter 6, where it is stated that HIV/AIDS constitutes a major national health problem in Uganda, with high prevalence rates and high number of HIV-infected pregnant women in need of antiretroviral drugs to prevent mother-to- child transmission. With nearly 80% of those infected being between the ages of 15 and 45, the disease takes away the breadwinners of families, thereby causing a huge impact on people's lives in this country. The findings of this part showed that key determinants of HIV transmission were closely related to patterns of risky attitudes and socio-demographic characteristics (e.g., lack of education and general awareness of health problems) of certain subgroups of the population surveyed, highlighting the importance of behavioral changes, public awareness, and health education in reducing the risk of getting HIV/AIDS. It was also suggested that the relationship between age and condom use should be taken into account when prevention measures are developed in Uganda. Apparently, this country needs sufficient resources to establish effective surveillance programs, with emphasis placed on evaluating the current trends in HIV transmission, and embarking on nationwide prevention and education programs to enhance public health planning. The analysis presented in this part represents a necessary step towards further, in-depth research that is likely to require the use of innovative research methods, such as new approaches to data collection (e.g., interactive voice recognition technology, and audio computer-assisted self-interviewing technology) to reduce bias and increase power and precision. Designed based on body of knowledge provided by some of the most experienced professionals in the field, this book covers:
Study designs used for the purpose of each database; Descriptive statistics, i.e., characteristics of study samples;
Bivariate analyses, i.e., Chi-square tests, T-tests, and ANOVAs; Multivariate analyses, i.e., MANOVAs; Numerous examples of ready-to-use research questions and null and alternative hypotheses; Inclusion of covariates, i.e., ANCOVA and MANCOVA, in statistical tests; How to report the results; and Two detailed discussion sections (one on each database), parts of which can be used as templates.
In order to better interpret the results of the analyses carried out, a range of tests that are necessary to support the findings are performed. All statistical tests are two-tailed, and p < .05 is considered significant. The overall conclusions for the analyses are summarized in the final chapter.
The main objective of this book is to provide a practical, step-by-step guide to assist those who are or will be practicing medicine, but lack basic understanding of statistics. In particular, it aimed to pay special attention to aspects identified as the ones Masters and PhD students struggle with the most. It focuses on laying the foundation for developing high level statistical skills and abilities required to evaluate appropriately and accurately scientific information and data on public health issues.
Five chapters of the book aim to examine the effects of three noise conditions on students' performance while taking two tests. Several other chapters examine the Ugandan database and identify which variables can be analyzed in order to distinguish which specific subgroups should be targeted for the development of a comprehensive regional HIV/AIDS intervention or a suitable education program. In addition to that, the book also attempts to:
1. Assess the strengths and weaknesses of different quasi-experimental research methods to control internal validity threats;
2. Use SPSS to perform ANOVAs (One-Way and Two-Way ANOVAs, including a 3x2 Factorial) to analyze the differences between the means of two groups or more;
3. Interpret the SPSS outputs and explain the importance of selecting appropriate statistical analyses to test the proposed hypotheses that are used to specifically focus on the purpose of the studies under investigation; and
4. Explain the differences and similarities between the MANOVA, ANCOVA and the ANOVAs.
Recently, much scientific interest has been converging on the effect of background classical music (Mozart or Chopin) on students' ability to learn and perform better on exams. The study presented in the first part of the book attempts to determine if students perform better on examination in math and reading when they are exposed to classical music (Chopin or Mozart). It is anticipated that such evaluation will provide knowledge on whether listening to music disrupt or help concentration and performance on tests of reading and math skills. A total of 300 students participated in this study, and the inclusion criteria were sixth graders who had tested at grade level or higher in both reading and math on their last SAT 9 or other test as required by the State where the experiment was conducted. The subjects were randomly assigned to three groups, each of which had 50 males and 50 females. The testing conditions were as follows:
1. Classical Music played in background at a level which is 5 dBA over ambient levels;
2. Traffic noise played in background at a level of 60 dBA; and
3. Quiet testing room which met all recommended standards for testing conditions.
The students took one examination under only one of the three testing conditions (Math or reading), and then took the second examination after a 45 minute break.
This study can be described as a 3x2 factorial design, with three type of noise (classical, traffic, or quiet) and two examinations (math or reading). To be more precise, it is a randomized, three-group experimental design in which students were randomly assigned to one of three groups (100 subjects each):
1. Condition 1(classical music);
2. Condition 2 (traffic noise); and
3. Condition 3 (quiet testing room).
Each experimental group was further subdivided into two counterbalancing groups with 50 subjects each. This means that only 100 students were tested per noise testing condition. By using the design notation (Abrahams, n.d.), this study design can be depicted as follows:
Abbildung in dieser Leseprobe nicht enthalten
where R represent a group which was randomly assigned, O1 stands for a Math
achievement test, O_{2} stands for a reading achievement test, and Xc1, Xc2, and Xc3 are used to depict each of the three testing conditions, respectively.
Given the variables measured for this study, the planned analyses consisted of descriptive statistics followed by comparisons of mean math and reading test scores between the three testing conditions, using ANOVAs, ANCOVA, and MANOVA. The overall research question is as follows: Under which noise testing conditions did the students scored the highest grades?
Of the 300 students participating in this analysis, the vast majority were White and Hispanic (Whites 38.7%, Hispanics 31.7%, Afro-Americans 10.7%, American Indians 2.7%, Asian-Americans 11.3%, and Pacific Islanders 5%). The household incomes of participants varied widely, from $20,001 to $100,000, with the mode income (i.e., the income happening most frequently) being $50,001 - $60,000. Likewise, the parents' highest level of education completed differed significantly among students. While 21.3% of the students' parents had completed two years of college, the percentage of parents who did not complete high school was as low as 9.3%. Also, only 4.3% of the parents had earned postgraduate degrees, whereas 14% had completed high schools. The students' minimum reading test score was 13, while the maximum reading score was 22 (M = 17.11, SD = 1.87). In contrast, the students' minimum math test score was 11 compared to a maximum score of 21 (M = 17.00, SD = 2.36).
The following research questions and hypotheses were formulated to be answered by bivariate analyses:
Research question 1. Is there a significant difference between type of noise and reading test scores among the tested students?
H0: There is no significant difference between type of noise and reading test scores among the tested students (μ1=μ2=μ3).
HA: There is a significant difference between type of noise and reading test scores among the tested students (μ1≠μ2≠μ3).
By replacing "reading test scores" with "math test scores" as the dependent variable, the following research questions and hypotheses emerge:
Research question 2. Is there a significant difference between type of noise and math test scores among the tested students?
H0: There is no significant difference between type of noise and math test scores among the tested students (μ1=μ2=μ3).
HA: There is a significant difference between type of noise and math test scores among the tested students (μ1≠μ2≠μ3).
Nevertheless, the following joint research question and hypotheses can be used instead of the above mentioned ones:
Joint research question. Which type of noise (classical, traffic, or quiet) would result in the greatest test scores among the tested students?
H0: There will be no significant difference in the mean test scores for the different levels of type of noise (μ1=μ2=μ3).
HA: There will be a significant difference in the mean test scores for the different levels of type of noise (μ1≠μ2≠μ3).
Results. Independent samples t -tests were conducted to evaluate the hypothesis that students who took the test under classical music (Mozart) testing condition would have higher test scores compared to the two other conditions. The results from the statistical analyses were as follows:
1) There was a significant difference in reading test scores between students who took the test under classical music (Mozart) testing condition (M = 18.05, SD = 1.80) compared to traffic noise testing condition (M = 15.68, SD = 1.37), t (184.869) = 10.478, p < .001; Levenes < .001.
2) There was no significant difference in reading test scores between students who took the test under classical music (Mozart) condition (M = 18.05, SD = 1.80) compared to the quiet-standard condition (M = 17.61, SD = 1.48), t (191.018) = 1.887, p = .061; Levenes = .002.
3) Students' reading test scores were significantly higher under quiet-standard condition (M = 17.61, SD = 1.48) compared to traffic noise condition (M = 15.68, SD = 1.37), t (198) = - 9.559, p < .001.
4) There was a significant difference in math test scores between students who took the test under classical music (Mozart) testing condition (M = 18.42, SD = 1.79) compared to quiet-standard testing condition (M = 17.83, SD = 1.54), t (198) = 2.498, p = .013.
5) The students' math test scores were significantly higher under classical music (Mozart) testing condition (M = 18.42, SD = 1.79) than under traffic noise condition (M = 14.76, SD = 1.84), t (198) = 14.233, p < .001.
6) The students' math test scores were significantly higher under quiet-standard condition (M = 17.83, SD = 1.54) than under traffic noise condition (M = 14.76, SD = 1.84), t (198) = -12.790, p < .001.
In addition to t -tests, Chi-Square tests were used to answer the following research questions and hypotheses:
Research question 1. Is there a relationship between type of noise and ethnicity of the tested students?
H0: Type of noise and ethnicity of the tested students are independent (i.e., there is no relationship between these two nominal variables).
HA: Type of noise and ethnicity of the tested students are dependent (i.e., there is a relationship between these two nominal variables).
Research question 2. Is there a relationship between type of noise and parental highest level of education completed of the tested students?
H0: Type of noise and parental highest level of education completed are independent. HA: Type of noise and parental highest level of education completed are dependent.
Results. Chi-Square tests were employed to test whether type of noise was independent of ethnicity and parental highest level of education. The results suggested that the ethnicity of the tested students was independent of the testing conditions applied in this study. In other words, the test was not significant, χ^{2} (10, N = 300) = 16.824, p = .078.
For parents' highest level of education completed, the test resulted in rejecting the null hypothesis for parental highest level of education in favor of the alternative hypothesis. This indicates that parents highest level of education and type of noise are not independent (i.e., they are dependent). To be more precise, there is a relationship between type of noise and parents highest level of education, χ^{2} (16, N = 300) = 71.982, p < .001. However, since nine cells (33.3%) had expected count less than 5, the parents level of education combined variable, named ParentEd1, was created by recoding postgraduate, master's degree, and doctorate degree as postgraduate and higher. A new Chi-Square test was then carried out with ParentEd1 and Noisetype, and this test was still very significant, χ^{2} (12, N = 300) = 67.315, p < .001, and the assumption was not violated in this case.
Interpretation. As it can be seen from Table 1, the Chi-Square test for parents' level of education was significant because significantly more students who were tested under the traffic noise condition had parents who did not complete high school (21 students of a total of 100) than expected (9.3 students). In other words, this cell was overrepresented.
The value for this particular cell = [(21-9.3)^{2} /9.3] = 14.72. Since χ^{2} value was 67.315, thus this cell accounted for 14.72/67.315 = 21.87%.
In addition, additional cells that made this test statistically significant can also be found in Table 1. Apparently, (a) no students who were tested under the quiet-standard condition had parents who did not complete high school (0 students of a total of 100) than expected (9.3 students); and (b) significantly more students who were tested had parents who had completed high school (25 students of a total of 100) than expected (14 students). This means that the first mentioned cell was underrepresented and the second was overrepresented in the actual sample.
The value for these two cells = [(0-9.3)^{2} /9.3] + [(25-14)^{2} /14] = 9.30 + 8.64 = 17.94. Since χ^{2} value was 67.315, thus these cells accounted for 17.94/67.315 = 26.65%. Overall, all the above-mentioned cells accounted for 48.52%.
Table 1. Crosstabulation of Parents' Highest Level of Education by Type of Noise
Abbildung in dieser Leseprobe nicht enthalten
χ^{2} (12, N = 300) = 67.315, p < .001
After performing several bivariate tests, one-way and two-way ANOVAs were used to test the theory that classical music may increase students' performance. In this connection, several separate ANOVAs were performed. Before presenting them in detail, here is an overview of the results of testing the assumptions of ANOVA.
Two major assumptions of ANOVA are (a) the normality of the dependent variable (i.e., the dependent scaled variable should be normally distributed within groups); (b) homogeneity of variances (i.e., it is assumed that the variances of the dependent variable is the same across the different groups being studied) (StatSoft, 2010; Cramer & Howitt, 2004; Field, 2005; Chernick & Friis, 2003). Overall, the F statistic is quite robust against violation of the homogeneity of variances assumption, especially when the sample sizes are equal (Field, 2005). The results of testing this assumption are presented in the next sections. With regards to the normality assumption, it must be highlighted that the ANOVA test is also robust to deviations from normality provided the sample size is large (StatSoft, 2010).
Further, the skewness of the distribution usually does not have a sizable effect on the F statistic, while the kurtosis does affect the F statistic and is associated with making both Type I and Type II errors (StatSoft, 2010). If the kurtosis is greater than 0, the F statistic tends to be too small and the null hypothesis cannot be rejected even though it is false (i.e., Type II error) (StatSoft, 2010). In contrast, if the kurtosis is less than 0, the F statistic tends to be too large, which increases the risk of committing Type I error (i.e., rejecting the null hypothesis even though it is true) (StatSoft, 2010).
In order to explore the distributions of the dependent variables in this database, the "Frequencies" command was used and the results are illustrated in Table 2. Noticeably, the assumption of normality was almost met for math test score, whereas there was some evidence of non-normality for the reading test score. To be more precise, the mean value of reading test score deviated slightly from its respective median value, while the mean math scores equaled the median, but the mode was less than the mean and median. Further, from Table 2, twice the standard error of skewness for both reading and math test scores would be 2 x 0.141= 0.282. By looking at the range from - 0.282 to + 0.282 and checking whether the values of Skewness for these two variables fall within this range, it becomes clear that the distribution of math test score is significantly negatively skewed (-0.717), while that of reading test score is slightly negatively skewed (Price, 2000; USouthAl, n.d.).
Table 2. Descriptive Statistics for Reading and Math Test Scores
Abbildung in dieser Leseprobe nicht enthalten
In addition to that, another descriptive statistic that can be used to test the normality of the variable is kurtosis. From Table 2, twice the standard error of kurtosis for reading test scores and math test scores would be 2 x 0.281 = 0.562. Since the values of kurtosis for reading test scores (-0.258) and math test scores (0.053) fall within the range from - 0.562 to + 0.562, the data may be considered as normally distributed by this measure (Price, 2000). Nonetheless, it must be noted that due to the relatively small sample size, any deviations from normality may have effect on the outcome of ANOVA tests because, according to the central limit theorem, only with a large sample size, the sampling distribution of means will be normally distributed (Cramer & Howitt, 2004). Therefore, the results will be interpreted with caution.
Effects of Parents' Highest Level of Education on Test Scores
Research questions and corresponding hypotheses. The following research question and hypotheses are proposed to examine the effects of parents' highest level of education on students' reading and math test scores:
Research question 1. Is there a significant difference in the average reading test score among the students tested for the different levels of parents' highest level of education completed?
H0: There is no significant difference in the average reading test score among the students tested for the different levels of parents' highest level of education completed (μ1 = μ2 = μ3 = μ4 = μ5 = μ6 = μ7 = μ8 = μ9).
HA: There is a significant difference in the average reading test score among the students tested for the different levels of parents' highest level of education completed (μ1 ≠ μ2 ≠ μ3 ≠ μ4 ≠ μ5 ≠ μ6 ≠ μ7 ≠ μ8 ≠ μ9).
Research question 2. Is there a significant difference in the average math test score among the students tested for the different levels of parents' highest level of education completed?
H0: There is no significant difference in the average math test score among the students tested for the different levels of parents' highest level of education completed (μ1 = μ2 = μ3 = μ4 = μ5 = μ6 = μ7 = μ8 = μ9).
HA: There is a significant difference in the average math test score among the students tested for the different levels of parents' highest level of education completed (μ1 ≠ μ2 ≠ μ3 ≠ μ4 ≠ μ5 ≠ μ6 ≠ μ7 ≠ μ8 ≠ μ9).
Results. A one-way between subjects ANOVA examined the effect of parents' highest level of education completed on the students' reading test score. The test for homogeneity of variance was not significant (Levene's statistic F (6, 293) = 1.064, p = .384), indicating equal variances.
Overall, there was a significant difference in reading test scores among students for the different levels of parents' highest level of education, F (6, 293) = 4.106, p = .001. A Bonferroni post hoc comparison indicated that students whose parents did not complete high school (M = 15.89, SD = 1.75) scored lower than those whose parents had an associate degree or 2 years of college (M =17.47, SD =1.86), Bonferroni = .003, and those whose parents had 3-4 years of college with no degree (M =17.55, SD =1.54), Bonferroni = .002 (see Table 3). Also, students whose parents had completed postgraduate studies or had master's or doctorate degrees (M = 17.94, SD = 1.35) scored significantly higher than those whose parents did not complete high school (M = 15.89, SD = 1.75), Bonferroni = .004.
Table 3. Post Hoc Tests (Bonferroni) for the Effects of Parent' Highest Level of Education Completed on Students' Reading Test Scores
Abbildung in dieser Leseprobe nicht enthalten
*. The mean difference is significant at the .05 level.
Similarly, another one-way between subjects ANOVA tested the effect of parents' highest level of education on students' math test scores. The test for homogeneity of variance was not significant (Levene's statistic F (6, 293) = 0.859, p = .525), indicating equal variances. Overall, there was a significant difference in math test scores among students for the different levels of parents' highest level of education, F (6, 293) = 7.661, p < .001. Moreover, a Bonferroni post hoc comparison revealed that students whose parents had an associate degree or 2 years of college (M =17.50, SD = 2.25), scored significantly higher in math test than those whose parents did not complete high school (M = 15.79, SD = 2.20), Bonferroni = .015, and those whose parents had completed high school (M = 15.83, SD = 2.08), Bonferroni = .004, or even those whose parents had completed one year of college or trade school (M = 16.23, SD = 2.50), Bonferroni = .045 (see Table 4).
Table 4. Post Hoc Tests (Bonferroni) for the Effects of Parent' Highest Level of Education Completed on Students' Math Test Scores
Abbildung in dieser Leseprobe nicht enthalten
*. The mean difference is significant at the .05 level.
Additionally, students whose parents had completed 3-4 years of college with no degree (M = 17.68, SD = 2.00) scored significantly higher in math exam than those whose parents did not complete high school (M = 15.79, SD = 2.20), Bonferroni = .006, and those whose parents had completed high school (M = 15.83, SD = 2.08), Bonferroni = .001, or even those whose parents had completed one year of college or trade school (M = 16.23, SD = 2.50), Bonferroni = .015. Also, students whose parents had earned a BS/BA degree (M = 17.92, SD = 2.00) scored significantly higher in math exam than those whose parents did not complete high school (M = 15.79, SD = 2.20), Bonferroni = .003, and those whose parents had completed high school (M = 15.83, SD = 2.08), Bonferroni = .001, or even those whose parents had completed one year of college or trade school (M = 16.23, SD = 2.50), Bonferroni = .007. Finally, students whose parents had completed postgraduate studies or Master's or Doctorate degrees (M = 18.06, SD = 2.53) scored significantly higher in math test than those whose parents did not complete high school (M = 15.79, SD = 2.20), Bonferroni = .016, and those whose parents had completed high school (M = 15.83, SD = 2.08), Bonferroni = .009.
Effects of Gender and Type of Noise on Reading Test Scores
Apart from that, a two-way within subjects ANOVA was conducted to compare the effect of gender and type of noise on students' performance.
Research question and corresponding hypotheses. The following research question and hypotheses are proposed:
Research question: Is there a significant difference between the effects of type of noise and gender on students' reading test scores?
- H01: There are no significant differences between the effects of type of noise on students' reading test scores.
- HA1: There are significant differences between the effects of type of noise on students' reading test scores.
- H02: There are no significant differences between reading test scores of male and female students.
- HA2: There are significant differences between reading test scores of male and female students.
- H03: There is no significant interaction between the effects of type of noise and gender.
- HA3: There is a significant interaction between the effects of type of noise and gender.
Results. The test for homogeneity of variance was significant (Levene's statistic F (5, 294) = 6.494, p < .001), implying that error variance of the dependent variable is not equal across groups. Overall, there was a significant difference in reading test scores for type of noise and gender, F (5, 294) = 27.874, p < .001 (see Table 5). There was a significant main effect of type of noise on reading test scores, F (2, 294) = 66.087, p < .001, whereas the main effect of gender was not significant, F (1, 294) = 0.798, p = .372. However, the interaction effect was significant, F (2, 294) = 3.198, p = .042.
The effect size ( 2 = 0.322) was large (see Table 5), the power to detect a significant difference in the sample was very high (power = 1.00), and 32% of the variance in the dependent variable was accounted for by this model.
Table 5. ANOVA for the Effects of Type of Noise and Gender on the Students' Reading Scores
Abbildung in dieser Leseprobe nicht enthalten
The results of post hoc tests reported in Table 6 indicate that the average reading test scores was significantly lower in the traffic noise testing condition (M = 15.68, SD = 1.37) as compared to those scored in the classical music condition (M = 18.05, SD = 1.80), and those in the quiet-standard testing condition (M = 17.61, SD = 1.48), Bonferroni < .001.
Table 6. Post Hoc Tests (Bonferroni) for the Effects of Gender and Type of Noise on Students' Reading Test Scores
Abbildung in dieser Leseprobe nicht enthalten
*. The mean difference is significant at the .05 level.
Figure 3 shows the estimated marginal means of reading test scores as a function of type of noise and gender. Apparently, students scored higher under the classical music condition than they did under the other two testing conditions.
Abbildung in dieser Leseprobe nicht enthalten
Figure 1. Estimated marginal means of reading test scores for students tested by type of noise
and gender showing that type of background noise had a substantial effect on performance:
Students scored higher under the classical music (Mozart or Chopin) condition compared to both the traffic noise and the quiet-standard conditions, and that students scored best under the classical music condition and worst under the traffic noise condition.
Effects of Gender and Type of Noise on Math Test Scores
A last two-way within subjects ANOVA was conducted to compare the effect of gender and type of noise on students' math test scores.
Research question and corresponding hypotheses. The following research question and hypotheses are put forward:
Research question: Is there a significant difference between the effects of type of noise and gender on students' math test scores?
- H01: There are no significant differences between the effects of type of noise on students' math test scores.
- HA1: There are significant differences between the effects of type of noise on students' math test scores.
- H02: There are no significant differences between math test scores of male and female students.
- HA2: There are significant differences between math test scores of male and female students.
- H03: There is no significant interaction between the effects of type of noise and gender.
- HA3: There is a significant interaction between the effects of type of noise and gender.
Results. Testing the assumptions for ANOVA revealed that the test for homogeneity of variance was significant (Levene's statistic F (5, 294) = 10.491, p < .001), suggesting that error variance of the dependent variable is not equal across groups.
All effects were statistically significant at the .05 significance level (see Table 7).
[...]
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