Why should the analysis of variance?
- One-sample, two-sample t-test its ultimate goal is to analyze the two groups whether significant differences exist between data, but if you want to analyze multiple sets whether significant differences exist between the data is very difficult, so analysis of variance to solve this problem;
for example: t-test to analyze differences in school achievement in a class of men and women; and variance analysis can analyze a class of students from various provinces and cities enrollment results. - In the variance analysis, involving the control variables and random variables and observed variables;
for example: whether the amount of fertilizer crop production will bring a significant impact; Here, the control variables: the amount of fertilizer, the observed variables: crop yields, random variables: weather, temperature ......
Univariate analysis
Objective: To analyze the sample mean multiple sets of factors under the influence of a single control whether significant differences exist.
Conditions:
- Normality of the dependent variable at each level should be a normal distribution;
- Homoscedasticity, between the groups have the same variance;
- Independence between the groups are independent of each other.
case analysis:
Case Description: In a company, analyze the impact of advertising on sales. (Source: "Statistical Analysis with SPSS applications" (Fifth Edition) Xue Wei Chapter VI)
Topic analysis: in the title, not to forms of advertising are two, no way to use two independent samples t-test significant differences between the forms and sales analysis, at the same time, only one control factor, so the use of analysis of variance in univariate analysis .
The null hypothesis is proposed: there is no significant difference between advertising and sales.
Interface Procedure: - A Comparison of Mean - way ANOVA
Screenshot key steps:
Listing and dependent variables distinguish factors; factors: the control variable, the dependent variable list: observed variables
Result analysis:
| ANOVA |
|||||
| Sales |
|||||
|
|
sum of squares |
df |
Mean Square |
F |
Significance |
| Between groups |
5866.083 |
3 |
1955.361 |
13.483 |
.000 |
| s |
20303.222 |
140 |
145.023 |
|
|
| total |
26169.306 |
143 |
|
|
|
- And squares: groups sum of squares (SSA) is deteriorated caused by different levels of the controlled variable deviations within groups sum of squares (SSE) is caused by different levels of deterioration of random variables;
- df: degrees of freedom between the two groups, divided into four groups according to the present problem of the different forms of advertising, the degree of freedom k-1 = 4-1 = 3; the degree of freedom in the group nk = 144-k = 140;
- Mean Square: is the variance;
- F = SSA / (k-1) ÷ (SSE / (nk)) = within groups variance / class variance, F value is significantly greater than 1, illustrate the effect of the control variables on observed variables is larger than the random variable, whereas effective;
- P- value = 0.00 <0.05, the null hypothesis is rejected, that the different forms of advertising and sales areas on average had a significant influence on the sales Effects of different forms of advertising, not all zeros region.
Univariate analysis further examination:
Homogeneity of variance test: In the above-mentioned description, the single factor analysis satisfy the conditions of the respective groups to the variance is the same, it is necessary to verify the homogeneity of variance;
Ideas:
- The null hypothesis is proposed: the variance of each of the groups did not differ significantly;
- The method of using Levene F test between the lines;
- Ɑ p-value, and determines whether or not the same as the variance between groups.
Continue to analyze the above topics:
First, assuming the same different forms of advertising overall variance;
In the above screenshot was like interface, click "Options", get shown:
Select the homogeneity of variance test
Result analysis:
| description |
||||||||
| Sales |
||||||||
|
|
N |
Means |
Standard deviation |
SE |
Mean 95% confidence interval |
Minimum |
maximum |
|
| Lower limit |
Upper limit |
|||||||
| newspaper |
36 |
73.2222 |
9.73392 |
1.62232 |
69.9287 |
76.5157 |
54.00 |
94.00 |
| broadcast |
36 |
70.8889 |
12.96760 |
2.16127 |
66.5013 |
75.2765 |
33.00 |
100.00 |
| Propaganda material |
36 |
56.5556 |
11.61881 |
1.93647 |
52.6243 |
60.4868 |
33.00 |
86.00 |
| Experience |
36 |
66.6111 |
13.49768 |
2.24961 |
62.0442 |
71.1781 |
37.00 |
87.00 |
| total |
144 |
66.8194 |
13.52783 |
1.12732 |
64.5911 |
69.0478 |
33.00 |
100.00 |
| Homogeneity of variance test |
|||
| Sales |
|||
| Levene statistics |
df1 |
df2 |
Significance |
| .765 |
3 |
140 |
.515 |