A researcher conducted a repeated measures study comparing three treatment conditions. Refer to attached images and tale to answer a to d. MeanStd. DeviationNTreatment I1.001.4145Treatment II5.002.3455Treatment III6.001.5815 In APA format, report the F-ratio related to the treatment effect:Is this treatment effect significant?What is the partial η2 value for the treatment effect?Is this a weak, moderate, or strong effect? ### Mauchly's Test of Sphericity**Measure:** MEASURE_1| Within Subjects Effect | Mauchly's W | Approx. Chi-Square | df | Sig. | Greenhouse-Geisser | Huynh-Feldt | Lower-bound ||------------------------|-------------|---------------------|----|-----|--------------------|-------------|-------------|| treatment | 0.360 | 3.065 | 2 | 0.216| 0.610 | 0.737 | 0.500 |**Explanation:**Mauchly's test is utilized to verify the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. This means it checks sphericity, which is the condition where variances of the differences between all combinations of related groups (levels) are equal.- **Design:** - Intercept - Within Subjects Design: treatment- **Notes:** - Mauchly's W is the test statistic. - Approx. Chi-Square indicates the Chi-Square approximation. - df stands for degrees of freedom. - Sig. indicates the p-value for the test. - Greenhouse-Geisser and Huynh-Feldt are used to adjust the degrees of freedom for the averaged tests of significance due to violations of sphericity. Corrected tests are found in the Tests of Within-Subjects Effects table. - Lower-bound provides the minimum possible value of epsilon, a measure related to the degree of non-sphericity. This test is crucial in ensuring the validity of within-subjects ANOVA results and corrections are applied when sphericity is violated. The table titled "Tests of Within-Subjects Effects" is related to the measure named "MEASURE_1." It provides statistical results for a repeated measures analysis of variance (ANOVA). The table is divided into two main sources: "treatment" and "Error(treatment)." Each source has several corrections applied for sphericity: Sphericity Assumed, Greenhouse-Geisser, Huynh-Feldt, and Lower-bound.### Explanation of Columns:- **Source**: Indicates the variable being analyzed. Two sources are noted: the main effect of "treatment" and its associated error. - **Type III Sum of Squares**: Measures the total variability for the source. It is constant (70.000) for treatment and (10.000) for error across all corrections. - **df (degrees of freedom)**: Indicates the number of independent values. For treatment, it varies depending on the correction method (2, 1.220, 1.474, 1.000), while for error, it remains at 8 across corrections. - **Mean Square**: Calculated by dividing the Type III Sum of Squares by df, representing an average amount of variability. Varies across methods, showing highest value for Lower-bound under treatment (70.000).- **F**: The F-ratio, which compares the model to the error, showing a value of 28.000 across all corrections for treatment. - **Sig. (Significance)**: The p-value indicating statistical significance, with values ranging between <.001 and .006 for treatment, suggesting significant effects. - **Partial Eta Squared**: Provides a measure of effect size. It is consistently .875 for treatment across methods, indicating a large effect size.This table helps assess the influence of the "treatment" factor, demonstrating significant results across different corrections for sphericity.

Introduction: It is required to report the F-ratio in APA format.

Answered: A researcher conducted a repeated…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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A researcher conducted a repeated measures study comparing three treatment conditions. Refer to attached images and tale to answer a to d.

	Mean	Std. Deviation	N
Treatment I	1.00	1.414	5
Treatment II	5.00	2.345	5
Treatment III	6.00	1.581	5

In APA format, report the F-ratio related to the treatment effect:

Is this treatment effect significant?

What is the partial η² value for the treatment effect?

Is this a weak, moderate, or strong effect?

### Mauchly's Test of Sphericity

**Measure:** MEASURE_1

| Within Subjects Effect | Mauchly's W | Approx. Chi-Square | df | Sig. | Greenhouse-Geisser | Huynh-Feldt | Lower-bound |
|------------------------|-------------|---------------------|----|-----|--------------------|-------------|-------------|
| treatment | 0.360 | 3.065 | 2 | 0.216| 0.610 | 0.737 | 0.500 |

**Explanation:**

Mauchly's test is utilized to verify the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. This means it checks sphericity, which is the condition where variances of the differences between all combinations of related groups (levels) are equal.

- **Design:**
- Intercept
- Within Subjects Design: treatment

- **Notes:**
- Mauchly's W is the test statistic.
- Approx. Chi-Square indicates the Chi-Square approximation.
- df stands for degrees of freedom.
- Sig. indicates the p-value for the test.
- Greenhouse-Geisser and Huynh-Feldt are used to adjust the degrees of freedom for the averaged tests of significance due to violations of sphericity. Corrected tests are found in the Tests of Within-Subjects Effects table.
- Lower-bound provides the minimum possible value of epsilon, a measure related to the degree of non-sphericity.

This test is crucial in ensuring the validity of within-subjects ANOVA results and corrections are applied when sphericity is violated.

The table titled "Tests of Within-Subjects Effects" is related to the measure named "MEASURE_1." It provides statistical results for a repeated measures analysis of variance (ANOVA). The table is divided into two main sources: "treatment" and "Error(treatment)." Each source has several corrections applied for sphericity: Sphericity Assumed, Greenhouse-Geisser, Huynh-Feldt, and Lower-bound.

### Explanation of Columns:

- **Source**: Indicates the variable being analyzed. Two sources are noted: the main effect of "treatment" and its associated error.

- **Type III Sum of Squares**: Measures the total variability for the source. It is constant (70.000) for treatment and (10.000) for error across all corrections.

- **df (degrees of freedom)**: Indicates the number of independent values. For treatment, it varies depending on the correction method (2, 1.220, 1.474, 1.000), while for error, it remains at 8 across corrections.

- **Mean Square**: Calculated by dividing the Type III Sum of Squares by df, representing an average amount of variability. Varies across methods, showing highest value for Lower-bound under treatment (70.000).

- **F**: The F-ratio, which compares the model to the error, showing a value of 28.000 across all corrections for treatment.

- **Sig. (Significance)**: The p-value indicating statistical significance, with values ranging between <.001 and .006 for treatment, suggesting significant effects.

- **Partial Eta Squared**: Provides a measure of effect size. It is consistently .875 for treatment across methods, indicating a large effect size.

This table helps assess the influence of the "treatment" factor, demonstrating significant results across different corrections for sphericity.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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