# Correlation and Simple Linear Regression

Correlation and simple linear regression methods assess the degree of strength, direction of association, and a linear summary of relationship existing between two variables, or observational units (Berg, 2004). In an effort to expose the descriptive analysis, correlational patterns resulting from the dataset DEL618_DHS618m1.sav, the writer/researcher hopes to examine the associative factors in light of the inferential statistics procedures that are paramount to the assignment. Such endeavor should help the writer/researcher to meet the goal of the theoretical basis for the assignment.

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Correlation and Linear Frameworks

The correlation and linear patterns usually found in statistical analyses indicate that the role of independent and dependent variables is essential in the analysis of data as well as the levels of measurement utilized. In bivariate statistics and regression, as Berg (2004), and Myers, Gamst, and Guarino (2006) asserted, a flexibility of roles of the variables: playing one role in one context, and another role in another context can help explain their effects based on data collection methods used. This is important for the type of research design, the writer/researcher posits.

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A linear regression shows how a distribution is presented depending on the values of a variable x, and how another variable y varies. The relationship between these variables is the key concern. There is an effort to define a best line to ascertain the paths of the measures of central tendency (mean, variance, standard deviation…) (Berg, 2004, p.24). A simple linear equation is defined as y = a+ bx, where y is defined as the dependent variable, and x as the independent variable. The intercept, a constant, is labeled as a, and b the coefficient, is also considered as a slope. A bivariate relationship captured in a scatterplot shows how the relationship, and the shape of the bivariate between the variables are presented.

Statistical Basis

The focus of this assignment is generated from a researcher’s willingness to examine factors influencing reading scores among school children. 7 variables considered yield substantive descriptive statistics showing whether correlation relationships exist among the variables. The descriptive statistics in tables 1, 2, 3 and 4 SPSS below provide a complete picture of the variables, frequencies generated, and guided the writer/researcher description of the results. The descriptive statistics show that females have a higher frequency percentage (55.6 %) than males (24.4%). Reading rank has a higher mean than visual acuity, a lower standard deviation, and variances compared to visual acuity. This seems to suggest that the predictor and outcome variables can be considered in a bivariate domain, and correlation designs, as one variable relate to another; but there are other missing factors and further analysis to substantiate a valid conclusion or result (Keppel, Saufley, & Tokunaga, 1992).

The mean, standard deviation, median, and variances for reading rank are: 16.43 (mean), standard deviation (6.188), median (18), and variance (38.287), compared to visual acuity of  10.41 (mean), standard deviation (6.254), median (10.00) median, and 39.114(variances). For every standard deviation increase in visual acuity, there will be a 6.188 standard deviation in reading rank values.

Furthermore, for visual acuity rank, with a standard deviation of SD =6.254, skewness of .121, as depicted in table 5, and reading rank with SD =6.188, and skewness of -.965, suggesting both a skewness < 3.3 (statistics/standard error of skewness), the distribution seems to be normal. In figure 1 showing the output of a scatterplot matrix for visual acuity, and reading rank, the writer/researcher sees enough linearity due to a possible relationship between the variables when also carrying the Pearson’s correlation. However, further considerations and analysis would be needed since the scatterplot may not be too linear.

Descriptive Statistics

Table 1. Descriptive Statistics. Selected Variables

Variables

Group

Mean

Standard Deviation

Median

Frequencies (%)

Variances

Gender

1.56

.502

2.00

.252

Female

30 (55.6)

Male

24 (44.4)

Total

54 (100)

ID

12.94

6.534

13.00

42.645

1

1(1.9)

2

1 (1.9)

3

1 (1.9)

4

4 (7.4)

5

1 (1.9)

6

1 (1.9)

7

4 (7.4)

8

4 (7.4)

9

4 (7.4)

10

1 (1.9)

11

1 (1.9)

12

14 (1.9)

13

4 (7.4)

14

4 (7.4)

15

4 (7.4)

16

1 (1.9)

17

1 (1.9)

18

1 (1.9)

19

4 (7.4)

20

4 (7.4)

21

1 (1.9)

22

1 (1.9)

23

1 (1.9)

24

4 (7.4)

Total

54 (100)

16.43

6.188

18.00

38.287

1

1 (1.9)

2

1 (1.9)

3

1 (1.9)

4

1 (1.9)

5

1 (1.9)

6

1 (1.9)

7

1 (1.9)

8

1 (1.9)

9

1 (1.9)

10

1 (1.9)

11

1 (1.9)

12

2 (3.7)

13

1 (1.9)

14

1 (1.9)

15

4 (7.4)

16

4 (7.4)

17

4 (7.4)

18

4 (7.4)

19

4 (7.4)

20

4 (7.4)

21

4 (7.4)

22

4 (7.4)

23

4 (7.4)

24

4 (7.4)

Total

54 (100)

Visual Acuity in Rank

10.41

6.254

10.00

39.114

1

4 (7.4)

2

4 (7.4)

3

1 (1.9)

4

1 (1.9)

5

4 (7.4)

6

4 (7.4)

7

4 (7.4)

8

1 (1.9)

9

1 (1.9)

10

5 (9.3)

11

4 (7.4)

12

1 (1.9)

13

1 (1.9)

14

1 (1.9)

15

5 (9.3)

16

1 (1.9)

17

1 (1.9)

18

1 (1.9)

19

5 (9.3)

20

5 (9.3)

Total

54 (100)

Table 2 Descriptive Statistics

N

Minimum

Maximum

Mean

Std. Deviation

Variance

Gender

54

1

2

1.56

.502

.252

Visual Acuity Rank

54

1

20

10.41

6.254

39.114

54

1

24

16.43

6.188

38.287

ID

54

1

24

12.94

6.534

42.695

Valid N (listwise)

54

Table 3.Visual Acuity Rank Frequencies

Frequency

Percent

Valid Percent

Cumulative Percent

Valid

1

4

7.4

7.4

7.4

2

4

7.4

7.4

14.8

3

1

1.9

1.9

16.7

4

1

1.9

1.9

18.5

5

4

7.4

7.4

25.9

6

4

7.4

7.4

33.3

7

4

7.4

7.4

40.7

8

1

1.9

1.9

42.6

9

1

1.9

1.9

44.4

10

5

9.3

9.3

53.7

11

4

7.4

7.4

61.1

12

1

1.9

1.9

63.0

13

1

1.9

1.9

64.8

14

1

1.9

1.9

66.7

15

5

9.3

9.3

75.9

16

1

1.9

1.9

77.8

17

1

1.9

1.9

79.6

18

1

1.9

1.9

81.5

19

5

9.3

9.3

90.7

20

5

9.3

9.3

100.0

Total

54

100.0

100.0

Frequency

Percent

Valid Percent

Cumulative Percent

Valid

1

1

1.9

1.9

1.9

2

1

1.9

1.9

3.7

3

1

1.9

1.9

5.6

4

1

1.9

1.9

7.4

5

1

1.9

1.9

9.3

6

1

1.9

1.9

11.1

7

1

1.9

1.9

13.0

8

1

1.9

1.9

14.8

9

1

1.9

1.9

16.7

11

1

1.9

1.9

18.5

12

2

3.7

3.7

22.2

13

1

1.9

1.9

24.1

14

1

1.9

1.9

25.9

15

4

7.4

7.4

33.3

16

4

7.4

7.4

40.7

17

4

7.4

7.4

48.1

18

4

7.4

7.4

55.6

19

4

7.4

7.4

63.0

20

4

7.4

7.4

70.4

21

4

7.4

7.4

77.8

22

4

7.4

7.4

85.2

23

4

7.4

7.4

92.6

24

4

7.4

7.4

100.0

Total

54

100.0

100.0

Table 5. Descriptive Statistics and Skewness

N

Minimum

Maximum

Mean

Std. Deviation

Variance

Skewness

Statistic

Statistic

Statistic

Statistic

Statistic

Statistic

Statistic

Std. Error

Gender

54

1

2

1.56

.502

.252

-.230

.325

Visual Acuity Rank

54

1

20

10.41

6.254

39.114

.121

.325

54

1

24

16.43

6.188

38.287

-.965

.325

ID

54

1

24

12.94

6.534

42.695

.066

.325

Valid N (listwise)

54

Figure 1. Scatterplot for visual Acuity and Reading Rank Matrix

Bivariate Statistic and Regression

Multiple Regression and P-values: a) Relationship Between Social Studies, Math,

and Reading. To ascertain the relationship between these variables, the writer/researcher

posits the following research questions, and hypotheses:

RQ1: What is the relationship between social studies, math, and reading considered concurrently?

Ho: There is no statistically significant relationship between social studies, math, and reading considered concurrently.

Halt: There is a statistically significant relationship between social studies, math, and reading considered concurrently.

SPSS correlations in table 1 results reveal a correlation of .342 between social studies and math, and .647 between social studies and reading. The analysis revealed a significant and positive correlation between math and reading. r=. 342, and p= .011 for math; and r=.647, with p = .000, p < 0.01 for reading. The null hypothesis is thus rejected for the alternative hypothesis.

Table 1. Correlations

Social Studies

Math

Social Studies

Pearson Correlation

1

.342*

.647**

Sig. (2-tailed)

.011

.000

N

54

54

54

Math

Pearson Correlation

.342*

1

.423**

Sig. (2-tailed)

.011

.001

N

54

54

54

Pearson Correlation

.647**

.423**

1

Sig. (2-tailed)

.000

.001

N

54

54

54

*. Correlation is significant at the 0.05 level (2-tailed).

**. Correlation is significant at the 0.01 level (2-tailed).

b) Partial correlation coefficients for variables reading and social studies with gender held as a constant.

RQ2: What is the relationship between reading and social studies with gender held constant?

Ho: There is no statisticallysignificant relationship between reading and social studies with gender held constant.

Halt: There is a statisticallysignificant relationship between reading and social studies with gender held constant.

A partial coefficient using bivariate regression analysis was conducted for the variable reading and social studies, holding the variable gender as constant. The analysis shows a positive correlation between social studies and reading r = .643, p= < .01. With a df (51), and p value < .01(statistically significant), the null hypothesis is rejected for the alternative hypothesis. Table 2 summarizes the analysis results.

Table 2 Partial Correlation Reading and Social Studies

Control Variables

Social Studies

Gender

Correlation

1.000

.643

Significance (2-tailed)

.

.000

df

0

51

Social Studies

Correlation

.643

1.000

Significance (2-tailed)

.000

.

df

51

0

c) Partial correlation coefficients for variables math and reading with gender held as a constant.

RQ3: Is there a relationship between math and reading with gender being held constant?

Ho: There is no statisticallysignificant relationship between math and reading with gender being held constant.

Ho: There is no statisticallysignificant relationship between math and reading with gender being held constant.

A SPSS analysis was conducted for the variables math and reading with gender held as a constant. The analysis showed a positive correlation with r =. 354, df (51), the correlation between math and reading was significant.p= .009 <0.01.

Table 3. Partial Correlation with Math and Reading

Control Variables

Math

Gender

Math

Correlation

1.000

.354

Significance (2-tailed)

.

.009

df

0

51

Correlation

.354

1.000

Significance (2-tailed)

.009

.

df

51

0

Appropriate Test to Measure Correlation Between Reading in Rank, and Visual

Acuity:  A bivariatetest would be appropriate to measure the correlation between reading in rank, and visual acuity. SPSS analysis revealed a negative correlation between reading rank and visual acuity. r = -0.067, p =0.628 < 0.01 is not statistically significant.

Table 4. Reading in Rank and Visual Acuity Correlations

Visual Acuity Rank

Pearson Correlation

1

-.067

Sig. (2-tailed)

.628

N

54

54

Visual Acuity Rank

Pearson Correlation

-.067

1

Sig. (2-tailed)

.628

N

54

54

Linear Regressions

1) Social Studies as a Predictor of Reading Scores: Consistent with the notion of

interactive effects of variables, espoused by Agresti (2011), linear regressions

summarizing relationships between variables ought to make sense of data, and seem to suggest more than one predictor is needed for generalizing regression analyses (Berk, 2004, p.21).

A bivariate regression analysis was conducted for social studies as a predictor for social studies. R2 =.419, F (1, 52) =37.449, reading = 55.347, Social studies = 4.257,

and p < 0.01 as statistically significant. Tables 6 to 8 summarize analysis results. Table 9 also shows the bivariate regression for social studies, and reading.

Table 5  Variables Entered/Removeda

Model

Variables Entered

Variables Removed

Method

1

Social Studiesb

.

Enter

b. All requested variables entered.

Table 6. Model Summary

Model

R

R Square

Std. Error of the Estimate

Change Statistics

R Square Change

F Change

df1

df2

Sig. F Change

1

.647a

.419

.407

11.113

.419

37.449

1

52

.000

a. Predictors: (Constant), Social Studies

Table 7ANOVAa

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

4624.798

1

4624.798

37.449

.000b

Residual

6421.739

52

123.495

Total

11046.537

53

b. Predictors: (Constant), Social Studies

Table 8 Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

95.0% Confidence Interval for B

Collinearity Statistics

B

Std. Error

Beta

Lower Bound

Upper Bound

Tolerance

VIF

1

(Constant)

55.347

7.597

7.285

.000

40.102

70.591

Social Studies

4.257

.696

.647

6.120

.000

2.861

5.652

1.000

1.000

Table 9 Bivariate Regression for Social Studies as Predictor of Reading

Regression Weights

Variables                                            b                   B

Social studies                                   4.257           .647

R2                         .419

F              37.449

2) Math as a Predictor of Reading Scores: A bivariate regression was conducted for math and reading scores. Tables 11, 12, and 13 summarized analysis results. The bivariate regressions regression model with social studies as a predictor shows:

R2= .179

R2= .163 (adjusted), F (1, 52) =11.326 is significantly significant p=.001, and reading=74.549+1.155(math).

Table 11 below shows, B=.423 for math; thus for every standard deviation increase in math scores there is a .423 standard deviation rise in reading scores.

Table 10. Bivariate Regression for Math as a Predictor of Reading Scores

Regression weighs

Variables                                 b                 B

Math                                   1.155            .423___________________________

Table 11 Model Summary

Model

R

R Square

Std. Error of the Estimate

Change Statistics

R Square Change

F Change

df1

df2

Sig. F Change

1

.423a

.179

.163

13.208

.179

11.326

1

52

.001

a. Predictors: (Constant), Math

Table 12 ANOVAa

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

1975.717

1

1975.717

11.326

.001b

Residual

9070.820

52

174.439

Total

11046.537

53

b. Predictors: (Constant), Math

Table 13 Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

95.0% Confidence Interval for B

Collinearity Statistics

B

Std. Error

Beta

Lower Bound

Upper Bound

Tolerance

VIF

1

(Constant)

74.549

8.036

9.277

.000

58.424

90.674

Math

1.155

.343

.423

3.365

.001

.466

1.844

1.000

1.000

The SPSS outputs are included for further review by the professor if needed.

References

Agresti, A. (2011). An introduction to categorical data analysis (2nd Ed). Hoboken, NJ: John Wiley & Sons.

Berk, R.A. (2004). Regression analysis. A constructive critique. Thousand Oaks, CA: Sage.

Keppel, G., Saufley, W.H., Jr., & Tokunaga, H. (1992). Introduction to design and analysis: A student’s handbook (2nd ed.). New York: W.H. Freeman.

Meyers, L. S., Gamst, G., & Guarino, A.J. (2006). Applied multivariate research. Design and interpretation. Thousand Oaks, CA: Sage.

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