BASIC STATISTICS REGRESSION ANALYSIS DISCRIMINANT ANALYSIS FACTOR ANALYSIS
BASIC STATISTICS
1. Paired t-tests for Brand Index Score
Table 1.1
Brand |
Brand Index Score Mean |
Standard Deviation |
Atkins |
27.1 |
4.1 |
Myoplex |
25.1 |
4.5 |
Snickers |
24.7 |
4.1 |
sample size = 62
The Brand Index score for each brand was calculated by summing ten Likert items pertaining to each brand. The highest Brand Index Score possible is 50, while the lowest is 10. Each of the three brands tested have a Brand Index Score in the middle, meaning respondents were neutral to each brand.
Table 1.2
Paired Samples |
t-ratio |
Atkins-Myoplex |
3.22* |
Myoplex-Snickers |
.564 |
Snickers-Atkins |
3.28* |
*p<=.15
The paired samples of Atkins and Myoplex and Snickers and Atkins are statistically significant, while the paired sample of Myoplex and Snickers is not. This means that in 85 or more samples drawn from the same population as this sample of 62 people, it would be expected that the difference between the brand index mean scores for Atkins and Myoplex, and Snickers and Atkins would be about the same as they are in this sample.
2. Between-Group t-tests
Table 2.1
Change Score |
Number of Respondents |
Brand Index Score Mean |
Standard Deviation |
t-ratio |
Up Movers |
29 |
27.4 |
3.8 |
.72 |
Down Movers |
29 |
26.7 |
4.6 |
Table 2.2
Change Score |
Number of Respondents |
Ad Index *Score Mean |
Standard Deviation |
t-ratio |
Up Movers |
29 |
7.6 |
2.5 |
.89 |
Down Movers |
29 |
7.1 |
2 |
*Ad Index Score was calculated by summing the positive responses to section 7 and 8 of the brand preference survey.
There are 29 individuals, 48.3% of the sample, whose attitude toward Atkins became more favorable following exposure to its advertisement. These individuals had a mean Brand Index Score of 27.4, and a mean Ad Index Score of 7.6. The results found in this sample are not statistically significant, meaning they could not be applied to a larger population.
At the same time 29 individuals’, 48.3% of the sample, attitude toward Atkins became less favorable after exposure to its advertisement. These individuals had a mean Brand Index Score of 26.7, and a mean Ad Index Score of 7.1. The results found in this sample are not statistically significant, meaning they could not be applied to a larger population.
3. Chi Squared Significance Test
Atkins Change Score |
|
Brand Index Score Above Median |
Brand Index Score Below Median |
Up Movers |
Count |
15 |
17 |
Row % |
46.4 |
53.6 |
Column % |
48.1 |
51.7 |
Total % |
23.2 |
26.8 |
Down Movers |
Count |
13 |
13 |
Row % |
50.0 |
50.0 |
Column % |
48.1 |
44.8 |
Total % |
23.2 |
23.2 |
Same |
Count |
1 |
1 |
Row % |
50.0 |
50.0 |
Column % |
3.7 |
3.4 |
Total % |
1.8 |
3.4 |
Chi-Squared=.072 *, Total Cell Count: 62
*p<=.15
The median Brand Index Score for Atkins is 28.0. Out of the 62 respondents 50% felt more favorably to Akins after viewing the advertisement, while 3.6% stayed the same, and 46.4% felt less favorable. Of the up movers 23.3% had a higher Brand Index Score than the mean for the sample, while 26.8% had a lower one. The down movers were spilt in comparison to the median Brand Index Score.
The Chi-Squared test is .072 meaning it is not statistically significant, and cannot be carried on to other samples.
4. Post vs. Pre Test Frequencies
Change Score |
Atkins |
Myoplex |
Snickers |
Up Movers |
29 |
27 |
28 |
Same |
6 |
4 |
11 |
Down Movers |
29 |
29 |
23 |
There are 29 individuals, 48.3% of the sample, whose attitude toward Atkins became more favorable following exposure to its advertisement. While, there were 48.3 % who felt less favorable.
There are 27 individuals, 43.5% of the sample, whose attitude toward Myoplex became more favorable following exposure to its advertisement. While, there were 46.8 % who felt less favorable.
There are 28 individuals, 45.2% of the sample, whose attitude toward Snickers became more favorable following exposure to its advertisement. While, there were 37.1 % who felt less favorable.
Any increase in favorability for each of the three brands was generally offset by an approximately equal group of respondents who found the brand less favorable. The effectiveness of the brand is inconclusive using pre vs. post testing numbers.
5. Frequency Counting Comparing Atkins and Myoplex
Out of the 62 respondents, 25 individuals or 40.3% rated Atkins as having a higher Brand Index Score than that of Myoplex. While 37 individuals or 59.7%, rated Myoplex as having a higher Brand Index Score than Atkins. This show that more respondents had a more favorable attitude toward Myoplex than Atkins.
6. Simple Correlation Coefficient
A simple correlation was calculated, comparing the Brand Index Score of Atkins to Myoplex. The coefficient correlation is .422, which means the Brand Index Score of Atkins and Myoplex are moderately correlated. This relationship is also statistically significant at an alpha level of .15. These means in 85 or more samples out of every 100 samples drawn from the same population as this sample of 62 people, it is expected that the correlation between the Brand Index Score for Atkins and Myoplex would be about the same as found in this sample. Therefore, we can project the results of the survey to the population.
7. Mean Comparison
Martial Status |
Mean Brand Index Score of Atkins |
Single |
28.5 |
Married |
26.9 |
Divorced |
21.8 |
Those respondents who said they were single had a mean Brand Index Score of 28.5 in regards to the Atkins brand. Married respondents had a mean of 26.9, while divorced people reported a mean of 21.8. While the differences in means is negligible single people seem to respond best to the Atkins advertisement.
TOP
REGRESSION ANALYSIS
This focuses on a regression analysis of the copy-testing results of three print brands for three different protein bars. The analysis was conducted using the pre-post ad exposure change score as the dependent variable, while the independent variables are the ten individuals Likert item responses. The sample size is 62.
|
Multiple Regression (R) |
Coefficient of Multiple Determinant (R2) |
Standard Error |
F-ratio |
Atkins |
.4 |
12.4% |
3.4 |
.70 |
Myoplex |
.5 |
21% |
3.2 |
1.27 |
Snickers |
.3 |
11% |
4.05 |
.58 |
|
Atkins |
|
Myoplex |
|
Snickers |
Likert Item (Brand Attitude) |
b |
beta |
t-ratio |
|
b |
beta |
t-ratio |
|
b |
beta |
t-ratio |
Constant |
0.6 |
|
0.16 |
|
2.0 |
|
0.79 |
|
1.1 |
|
0.29 |
It is a good protein bar. |
-0.3 |
-0.1 |
-0.85 |
|
-0.7 |
-0.3 |
-1.90 |
|
-0.6 |
-0.2 |
-1.47 |
It gives me energy. |
-0.1 |
0.0 |
-.017 |
|
0.2 |
0.1 |
0.37 |
|
-0.01 |
0.0 |
-0.11 |
It is a filling snack. |
0.1 |
0.0 |
.030 |
|
-0.5 |
-0.2 |
-1.14 |
|
0.2 |
0.1 |
0.44 |
I prefer it to other protein bars. |
-0.1 |
0.0 |
-0.14 |
|
0.7 |
0.3 |
2.01 |
|
0-.4 |
-0.1 |
-0.81 |
It helps me stay physically fit. |
0.6 |
0.3 |
1.32 |
|
-0.2 |
-0.1 |
-0.57 |
|
0.0 |
0.0 |
0.05 |
It helps me lose weight. |
0.1 |
0.0 |
0.16 |
|
.05 |
0.2 |
1.09 |
|
-0.1 |
0.0 |
-0.23 |
I do not like the taste of it |
-0.4 |
-0.2 |
-0.85 |
|
-0.1 |
0.0 |
-0.25 |
|
0.3 |
0.1 |
0.67
|
I would purchase it |
0.6 |
0.3 |
1.40 |
|
-0.1 |
-0.1 |
-0.43 |
|
-0.3 |
-0.1 |
-0.55 |
I believe athletes eat it. |
-0.7 |
-0.3 |
-1.50 |
|
-0.2 |
-0.1 |
-0.56 |
|
0.0 |
0.0 |
-0.09 |
It tastes good. |
-0.4 |
-0.2 |
-0.98 |
|
-0.2 |
-0.1 |
-0.67 |
|
0.4 |
0.1 |
0.98 |
Beta’s considered important => the highest beta/2
Atkins Regression
The multiple regression analysis for Atkins shows a low coefficient of multiple determination or R2. This means there is almost no relationship between how respondents rated the Atkins brand based on the ten-likert items and their brand attitude as indicated by the post-pre change score. Moreover, the standard error is 3.4, which means respondents were on average 3.4 units up or down from the regression line. This data is not significant, and therefore cannot be scaled to a larger population.
The betas that are considered important to this analysis are those related to five Likert items: It helps me stay fit, I do not like the taste of it, I would purchase it, I believe athletes eat it and it tastes good.
The un-standardized coefficients (b) show the relationship between the brand attitudes as understood by the Likert items, and how respondents favored the ads. They can be used to determine the equation of the regression line. The equation for Atkins regression line is: pre-post ad exposure change score = 0.6 - 0.3(its good) - 0.1(energy) + 0.1(filling) - 0.1(prefer) + 0.6(healthy) + 0.1(weight loss) – 0.4(bad taste) + 0.6(purchase) – 0.7(athletes) - 0.4(taste good).
Myoplex Regression
The multiple regression analysis for Myoplex shows a coefficient of multiple determination of 21%. This indicates that there is a small relationship between how respondents rated the Myoplex brand based on the ten Likert items and their brand attitude as understood by the post-pre change score. Furthermore, the standard error is 3.2, which means respondents vary from 3.2 above the regression line to 3.2 below. Moreover, the data is not statistically significant, therefore it cannot be applied to a larger population.
The betas that are considered important are: It’s a good protein bar, it is a filling snack, I prefer it to other protein bars, and it helps me lose weight.
Snickers Regression
The multiple regression analysis for Snickers shows a coefficient of multiple determination of 11%. This indicates a very weak relationship between the Likert items and the change score. Likewise, the snickers standard error is the highest of the three equaling 4.05. This means the actual results vary quite a bit from the regression line. Moreover, the data is not statistically significant, therefore it cannot be applied to a larger population.
The betas that are considered important are: It’s a good protein bar, it is a filling snack, and I prefer it to other protein bars.
Overall the analysis reflects poor relationships. The ten Likert items are poor predictors of the post-pre change exposure score.
TOP
DISCRIMINANT ANALYSIS
The discriminant analysis compares the ten Likert items and the pre-post change score for the Atkins brand. The independent variables are the ten Likert items, and the dependent variable is the change score.
Table 1: Discriminant Analysis
Likert Items |
Up movers |
Down Movers |
Standardized Discriminant Coefficient |
Un standardized Discriminant Coefficient |
Mean |
Standard Deviation |
Mean |
Standard Deviation |
It is a good protein bar. |
2.4 |
1.5 |
1.0 |
0.0 |
-.9 |
-.6 |
It gives me energy. |
1.8 |
1.3 |
1.5 |
.7 |
-.4 |
-.3 |
It is a filling snack. |
3.0 |
1.3 |
3.5 |
.7 |
-.4 |
-.3 |
I prefer it to other protein bars. |
3.3 |
1.1 |
3.5 |
.7 |
-.1 |
-.1 |
It helps me stay physically fit. |
2.8 |
1.8 |
3.0 |
2.8 |
1.0 |
.5 |
It helps me lose weight. |
2.8 |
1.1 |
2.5 |
.7 |
-.3 |
-.3 |
I do not like the taste of it |
2.4 |
1.2 |
2.0 |
1.4 |
0.0 |
0.0 |
I would purchase it |
3.1 |
1.6 |
3.5 |
2.1 |
1.0 |
.7 |
I believe athletes eat it. |
2.6 |
1.5 |
3.0 |
1.4 |
.1 |
.1 |
It tastes good. |
3.2 |
1.4 |
4.0 |
1.4 |
.1 |
0.0 |
Table 1 shows that the mean Likert score for the Up Movers as compared to the Down Movers is similar. The highest mean for the Up Movers is 3.3, while the highest for the Down Movers is 4.0. This helps further illustrate the parity between the two groups. However the standard deviation for the Down Movers appears to be smaller than that of the Up Movers. This means there is less variability among the Down Movers.
According the standard discriminant coefficient there are only three variables that are considered important. They are: Atkins is a good protein bar, it helps me stay physically fit, and I would purchase it. From a logical standpoint, the importance of these variables makes sense, since taste, and how healthy the protein bar is ultimately effect the decision to buy.
Table 2: Centroid Analysis
Group |
Score |
Up Movers |
-.1 |
Down Movers |
1.2 |
Table 3: Wilks Lambda
Wilks’ Lambda |
Chi-Square |
Degrees of Freedom |
.91 |
2.26 |
10 |
As indicated by Table 2, the group centroids of the up movers and down are not close to one another. This contributes to a Wilks’ Lambda of .91 and Chi-Square of 2.26. Both of these metrics are not statistically significant at the alpha level of .15; thus they cannot be applied to a larger sample.
Table 4: Confusion Matrix
|
Predicted up movers |
Predicted down movers |
Actual Up Movers |
29 |
0 |
Actual Down Movers |
2 |
0 |
T = (.935 – 0.5) / sqroot(((.935*.065) + (.5*.5))/ 31) = 4.34
As indicated by Table 4, 93.5% of actual respondents were classified correctly. At an alpha of .15 this data is statistically significant, therefore these findings can be applied to 85 or more samples drawn from the same population as this sample.
TOP
FACTOR ANALYSIS
The analysis provided below consists of a factor analysis, where the relationship for ten Likert items is examined. This analysis is conducted for all three brands of protein bars, Atkins, Myoplex, and Snickers. The sample size for this analysis consists of 60 respondents.
Table 1: Total Variance Explained
Factor |
Atkins |
Myoplex |
Snickers |
|
Eigenvalues |
Percent variance |
Cumulative variance |
Eigenvalues |
Percent variance |
Cumulative variance |
Eigenvalues |
Percent variance |
Cumulative variance |
I |
3.2 |
32.2 |
32.2 |
2.3 |
22.6 |
22.6 |
1.9 |
18.7 |
18.7 |
II |
1.6 |
17.0 |
49.1 |
1.6 |
16.0 |
38.6 |
1.6 |
16.1 |
34.8 |
III |
1.2 |
12.1 |
61.2 |
1.4 |
13.5 |
52.2 |
1.4 |
13.6 |
48.4 |
IV |
1.1 |
10.6 |
71.8 |
1.1 |
11.1 |
63.3 |
1.3 |
12.9 |
61.3 |
V |
.8 |
7.9 |
79.7 |
1.1 |
10.7 |
74.0 |
0.9 |
9.5 |
70.8 |
VI |
.6 |
6.4 |
86.1 |
0.8 |
7.9 |
81.9 |
0.8 |
8.3 |
79.2 |
VII |
.5 |
5.1 |
91.3 |
0.6 |
6.2 |
88.2 |
0.7 |
6.6 |
85.8 |
VIII |
.4 |
3.8 |
95.1 |
0.5 |
5.3 |
93.5 |
0.6 |
5.5 |
91.3 |
IX. |
.3 |
3.0 |
98.1 |
0.4 |
3.8 |
97.3 |
0.5 |
4.6 |
95.9 |
X |
.2 |
1.9 |
100 |
0.3 |
2.7 |
100.0 |
0.4 |
4.1 |
100.0 |
Four of ten factors for Atkins and Snickers have EigenValues above 1. This means that they explain the variance of a Likert item. For example factor III of Atkins explains about a 60% variance for the ten Likert items, leaving approximately 40% unexplained. While Myoplex has five factors that help explain the Likert items. Factor III of Myoplex has a cumulative variance of about 52%, meaning 48% of variance was left unexplained.
Table 2: Factor Varimax Rotated Matrix (Atkins)
Likert |
Communalities |
Factor matrix |
Varimax rotated matrix |
|
|
I |
II |
III |
IV |
I |
II |
III |
IV |
It is a good protein bar. |
0.7 |
-0.3 |
-0.5 |
0 |
0.6 |
0 |
0 |
-0.1 |
0.8 |
It gives me energy. |
0.7 |
-0.1 |
0.7 |
0.2 |
0.4 |
0.1 |
0.7 |
0.3 |
0 |
It is a filling snack. |
0.5 |
0.5 |
0.3 |
0.1 |
-0.4 |
0.3 |
0 |
0.1 |
-0.6 |
I prefer it to other protein bars. |
0.7 |
-0.3 |
-0.5 |
0.6 |
0 |
0.1 |
0.1 |
-0.7 |
0.3 |
It helps me stay physically fit. |
0.8 |
0.8 |
-0.4 |
-0.2 |
0.1 |
0.6 |
-0.6 |
0.2 |
0 |
It helps me lose weight. |
0.7 |
0.5 |
0.4 |
-0.3 |
0.5 |
0.3 |
0 |
0.8 |
0.1 |
I do not like the taste of it |
0.6 |
-0.7 |
0.2 |
-0.2 |
0 |
-0.7 |
0.2 |
0 |
0.1 |
I would purchase it |
0.8 |
-0.5 |
0.5 |
0.5 |
0.1 |
-0.1 |
0.8 |
-0.2 |
0 |
I believe athletes eat it. |
0.7 |
0.7 |
0 |
0.4 |
0.2 |
0.8 |
0 |
0.1 |
-0.1 |
It tastes good. |
0.6 |
0.7 |
0.1 |
0.4 |
0 |
0.7 |
0 |
0.1 |
-0.2 |
For the above table a Factor Matrix was used for further analysis because it has the least number of ambiguous rows. Contributing factors have values higher or equal to the absolute value of 0.5. The above data is heavily dispersed, but the majority of the Likert items are loaded in factor 1.
Table 3: Factor Varimax Rotated Matrix (Myoplex)
Likert |
Communalities |
Factor matrix |
|
Varimax rotated matrix |
|
|
|
I |
II |
III |
IV |
V |
I |
II |
III |
IV |
V |
It is a good protein bar. |
0.8 |
-0.6 |
-0.1 |
0.2 |
0.3 |
0.3 |
-0.3 |
0.3 |
0 |
0.6 |
0 |
It gives me energy. |
0.7 |
-0.4 |
0.4 |
0.5 |
-0.4 |
0.4 |
0.1 |
0.9 |
0.2 |
0.1 |
0.1 |
It is a filling snack. |
0.5 |
0.7 |
0.2 |
0.2 |
-0.3 |
0.1 |
0.7 |
0 |
-0.1 |
-0.4 |
0.2 |
I prefer it to other protein bars. |
0.2 |
-0.2 |
-0.3 |
0.2 |
0.6 |
0.4 |
0.1 |
0 |
0 |
0.8 |
0 |
It helps me stay physically fit. |
0.8 |
0.4 |
-0.2 |
0.7 |
0 |
-0.2 |
0.5 |
0.1 |
-0.7 |
0 |
-0.1 |
It helps me lose weight. |
0.8 |
0.6 |
0.5 |
0.3 |
0.3 |
0.2 |
0.8 |
-0.1 |
0.1 |
0 |
-0.3 |
I do not like the taste of it |
0.6 |
-0.1 |
0.8 |
-0.2 |
0.1 |
0.3 |
0.3 |
0.2 |
0.8 |
0 |
-0.1 |
I would purchase it |
0.2 |
0 |
-0.4 |
-0.3 |
-0.5 |
0.6 |
-0.1 |
0 |
0 |
0 |
0.9 |
I believe athletes eat it. |
0.3 |
0.6 |
0.1 |
-0.5 |
0.2 |
0.2 |
0.3 |
-0.6 |
0.3 |
-0.1 |
0.2 |
It tastes good. |
0.5 |
0.6 |
-0.5 |
0.1 |
0.1 |
0.3 |
0.4 |
-0.5 |
-0.4 |
0.2 |
0.4 |
The above table depicts both the factor matrix and varimax rotated matrix for Myoplex. It shows five factors. For further analysis the factor matrix was used. This is due to the fact that it has the most Likert items loaded into factor 1. However similarly to Atkins the data is fairly spread out.
Table 3: Factor Varimax Rotated Matrix (Snickers)
|
Communalities |
Factor matrix |
Varimax rotated matrix |
|
|
I |
II |
III |
IV |
I |
II |
III |
IV |
It is a good protein bar. |
0.8 |
0 |
0.4 |
-0.1 |
-0.7 |
-0.3 |
0 |
0.1 |
-0.7 |
It gives me energy. |
0.9 |
0.6 |
-0.4 |
-0.2 |
-0.2 |
0.8 |
0.1 |
-0.2 |
-0.2 |
It is a filling snack. |
0.5 |
0.4 |
-0.6 |
0 |
0.3 |
0.7 |
0 |
-0.1 |
0.3 |
I prefer it to other protein bars. |
0.6 |
-0.5 |
-0.4 |
0.4 |
-0.2 |
0 |
-0.4 |
0.7 |
0.1 |
It helps me stay physically fit. |
0.8 |
-0.2 |
0.7 |
-0.3 |
0.3 |
-0.7 |
0 |
-0.5 |
0 |
It helps me lose weight. |
0.5 |
-0.6 |
-0.2 |
-0.5 |
0 |
-0.1 |
-0.8 |
-0.2 |
0.1 |
I do not like the taste of it |
0.4 |
-0.3 |
-0.1 |
0.3 |
0.6 |
-0.2 |
0 |
0.1 |
0.7 |
I would purchase it |
0.5 |
0.6 |
0.2 |
0 |
-0.1 |
0.2 |
0.5 |
-0.3 |
-0.3 |
I believe athletes eat it. |
0.5 |
0.4 |
0.4 |
0.4 |
0.3 |
-0.1 |
0.7 |
-0.1 |
0.2 |
It tastes good. |
0.3 |
-0.2 |
0.1 |
0.7 |
-0.3 |
-0.2 |
0.2 |
0.8 |
0 |
The factor analysis for Snickers yielded less number of Likert items that are loaded into a particular factor. However, much like Atkins and Myoplex there are several factors causing the data to be spread out. Nonetheless, the factor matrix was used for further comparison because it has the least number of ambiguous rows.
Table 5 and 6: Attitude and Paired T-test
|
Mean |
N |
Std. Deviation |
ATKINS |
2.7 |
60.0 |
0.4 |
SNICKERS |
2.5 |
60.0 |
0.4 |
MYOPLEX |
2.5 |
60.0 |
0.5 |
|
MEAN DIFFERENCE |
STANDARD |
T |
ATKINS - MYOPLEX |
0.2 |
0.5 |
3.1* |
ATKINS - SNICKERS |
0.2 |
0.6 |
2.3* |
MYOPLEX - SNICKERS |
0.0 |
0.5 |
-0.4 |
*p < .15
Taking the average score of those Likert items that had a factor score of 0.5 or higher derived the above tables. The means for all the three brands are very close. Atkins had a mean brand score of 2.7, while both Snickers and Myoplex had brand scores of 2.5. Though Atkins had the highest score it was by a very narrow margin. However, the paired t-tests comparing Atkins with Myoplex and Atkins with Snickers are statistically significant at an alpha less than or equal to .15. Thus it can be concluded that in 85 or more samples out of every 100 samples taken from the same population as these 60 respondents, we would expect to find a difference of mean attitude scores of the same magnitude for each pair as shown in this sample.
TOP
HOME