Difference between revisions of "Manuals/calci/LEVENESTEST"

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<div style="font-size:30px">'''LEVENESTEST(xRange,ConfidenceLevel,LogicalValue)'''</div><br/>
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<div style="font-size:30px">'''LEVENESTEST (DataRange,ConfidenceLevel,NewTableFlag)'''</div><br/>
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*<math>xRange</math> is the set of values for the test.
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*<math>DataRange</math> is the set of values for the test.
 
*<math>ConfidenceLevel</math> is the value from 0 to 1.
 
*<math>ConfidenceLevel</math> is the value from 0 to 1.
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*<math>LogicalValue</math> is either TRUE or FALSE.
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*<math>NewTableFlag</math> is either TRUE or FALSE. TRUE for getting results in a new cube. FALSE will display results in the same cube.
  
 
==Description==
 
==Description==
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*Equal variances across samples is called homogeneity of variance or homoscedasticity.  
 
*Equal variances across samples is called homogeneity of variance or homoscedasticity.  
 
*To do the Levenes test we need the following assumptions:
 
*To do the Levenes test we need the following assumptions:
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   1.The Samples from the populations are independently of one another.  
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   1.The Samples from the populations are independent of one another.  
 
   2. The population under consideration are Normally Distributed.  
 
   2. The population under consideration are Normally Distributed.  
 
*For three or more variables the following statistical tests for homogeneity of variances are commonly used:
 
*For three or more variables the following statistical tests for homogeneity of variances are commonly used:
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*If the data surely is of normally distributed or nearly to normally distributed then  we can use the Bartlett test.
 
*If the data surely is of normally distributed or nearly to normally distributed then  we can use the Bartlett test.
 
*The Levene's test is defined as
 
*The Levene's test is defined as
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  <math>H_0=\sigma_1^2=\sigma_2^2=......=\sigma_t^2</math>.
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  <math>H_0 = \sigma_1^2 = \sigma_2^2=...... = \sigma_t^2</math>.
 
  <math>H_1</math>=Not all of the variances are equal.  
 
  <math>H_1</math>=Not all of the variances are equal.  
 
*Normally there are three versions of the Levenes test.  
 
*Normally there are three versions of the Levenes test.  
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*There are 1.Use of Mean.2.Use of Median.3.Use of  10% of Trimmed Mean.
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*There are  
 +
*1.Use of Mean.
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*2.Use of Median.
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*3.Use of  10% of Trimmed Mean.
 
*The Levene test statistic is:
 
*The Levene test statistic is:
 
<math>W=\frac{(N-k)\sum_{i=1}^k N_i(Z_i-Z)^2}{(k-1)\sum_{i=1}^k \sum_{i=1}^k \sum_{j=1}^{N_i} (Z_{ij}-Z_i)^2}</math>.
 
<math>W=\frac{(N-k)\sum_{i=1}^k N_i(Z_i-Z)^2}{(k-1)\sum_{i=1}^k \sum_{i=1}^k \sum_{j=1}^{N_i} (Z_{ij}-Z_i)^2}</math>.
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**<math>k</math> is the number of different groups to which the sampled cases belong.
 
**<math>k</math> is the number of different groups to which the sampled cases belong.
 
**<math>N</math> is the total number of cases in all groups.
 
**<math>N</math> is the total number of cases in all groups.
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**<math>N_i</math>  is the number of cases in the <math>i^th</math> group.
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**<math>N_i</math>  is the number of cases in the <math>i^{th}</math> group.
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**<math>Y_{ij} is the value of the measured variable for the <math>j_th</math> case from the <math>i^th</math> group.
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**<math>Y_{ij} is the value of the measured variable for the <math>j^{th}</math> case from the <math>i^{th}</math> group.
 
*Zij is satisfying the one of the following conditions:
 
*Zij is satisfying the one of the following conditions:
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*1.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the Mean of the <math>i^th</math> subgroup.
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*1.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the Mean of the <math>i^{th}</math> subgroup.
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*2.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the Median of the <math>i^th</math> subgroup
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*2.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the Median of the <math>i^{th}</math> subgroup
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*3.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the 10%Trimmed Mean of the <math>i^th</math> subgroup.
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*3.<math>z_{ij}=|y_{ij}-\bar{y_i}|</math>,Where <math>\bar{y_i}</math> is the 10%Trimmed Mean of the <math>i^{th}</math> subgroup.
 
*Levene's Testing Procedure:
 
*Levene's Testing Procedure:
 
*1. checking the assumptions.
 
*1. checking the assumptions.
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==Example==
 
==Example==
 +
{| class="wikitable"
 +
|-
 +
| X1 || X2
 +
|-
 +
| 3067 || 3200
 +
|-
 +
| 2730 || 2777
 +
|-
 +
| 2840 || 2623
 +
|-
 +
| 2913 || 3044
 +
|-
 +
| 2789 || 2834
 +
|}
 +
*=LEVENESTEST(B1:C5,.05,0)
 +
{| class="wikitable"
 +
|+LEVENES TEST
 +
|-
 +
!  !! DATA-0 !! DATA-1
 +
|-
 +
| Median || 2840 || 2834
 +
|-
 +
| Mean || 2867.8 || 2895.6
 +
|-
 +
| Variance || 16923.7 || 51713.3
 +
|-
 +
| Count || 5 || 5
 +
|-
 +
| df || 4 || 4
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|}
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 +
{| class="wikitable"
 +
|+SUMMARY OUTPUT
 +
|-
 +
! LEVENESTEST !! STATISTICS
 +
|-
 +
| W || 1.0439235110342522
 +
|-
 +
| F-Test || 0.38245649772919
 +
|-
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| a || 0.05
 +
|-
 +
| F || 0.32726010523405
 +
|-
 +
| p 1 & 2 Tail || 0.1524069466470822 || 0.3048138932941644
 +
|}
 +
 +
==Related Videos==
 +
 +
{{#ev:youtube|81Yi0cTuwzw|280|center|Levene's Test}}
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 +
==See Also==
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*[[Manuals/calci/SIGNTEST| SIGNTEST]]
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*[[Manuals/calci/FRIEDMANTEST| FRIEDMANTEST]]
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 +
 +
==References==
 +
 +
*[http://en.wikipedia.org/wiki/Levene%27s_test Levene's test documentation on Wikipedia]
 +
*[http://www.qimacros.com/hypothesis-testing/levenes-test/ Levene's test for variance in Excel]

Latest revision as of 15:57, 14 June 2018

LEVENESTEST (DataRange,ConfidenceLevel,NewTableFlag)


  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle DataRange} is the set of values for the test.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ConfidenceLevel} is the value from 0 to 1.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle NewTableFlag} is either TRUE or FALSE. TRUE for getting results in a new cube. FALSE will display results in the same cube.

Description

  • This function used to test the Homogeneity of variances.
  • Levene's test is used to test the Samples have equal variances.
  • Equal variances across samples is called homogeneity of variance or homoscedasticity.
  • To do the Levenes test we need the following assumptions:
  1.The Samples from the populations are independent of one another. 
  2. The population under consideration are Normally Distributed.
  • For three or more variables the following statistical tests for homogeneity of variances are commonly used:
   1.Levene's Test.
   2.Bartlett Test.
  • Levene's test is an alternative to the Bartlett test.
  • If the data surely is of normally distributed or nearly to normally distributed then we can use the Bartlett test.
  • The Levene's test is defined as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0 = \sigma_1^2 = \sigma_2^2=...... = \sigma_t^2}
.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1}
=Not all of the variances are equal.
  • Normally there are three versions of the Levenes test.
  • There are
  • 1.Use of Mean.
  • 2.Use of Median.
  • 3.Use of 10% of Trimmed Mean.
  • The Levene test statistic is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=\frac{(N-k)\sum_{i=1}^k N_i(Z_i-Z)^2}{(k-1)\sum_{i=1}^k \sum_{i=1}^k \sum_{j=1}^{N_i} (Z_{ij}-Z_i)^2}} .

    • where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is the result of the test.
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the number of different groups to which the sampled cases belong.
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the total number of cases in all groups.
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_i} is the number of cases in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} group.
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{ij} is the value of the measured variable for the <math>j^{th}} case from the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} group.
  • Zij is satisfying the one of the following conditions:
  • 1.Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{ij}=|y_{ij}-\bar{y_i}|} ,Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y_i}} is the Mean of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} subgroup.
  • 2.Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{ij}=|y_{ij}-\bar{y_i}|} ,Where is the Median of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} subgroup
  • 3.Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{ij}=|y_{ij}-\bar{y_i}|} ,Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y_i}} is the 10%Trimmed Mean of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} subgroup.
  • Levene's Testing Procedure:
  • 1. checking the assumptions.
  • 2.State the Null(H0) and alternative(H1) hypothesis.
  • 3.Decide on the Significance level (α).
  • 4.Finding the Critical value and Rejection Region.Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_1=t-1} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_2=N-t} .
  • 5.Compute the Levenes statistic using the formula.
  • 6.Then decision of the value of the test statistic,W is falls in the rejection region or if p-value ≤ α, then reject Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} .Otherwise, fail to reject Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} . For the computation p-value we have to use the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df_2} .
  • 7. Finally we have to conclude that the rejection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} or fail to rejection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} according to the test statistic at the significance level.

Example

X1 X2
3067 3200
2730 2777
2840 2623
2913 3044
2789 2834
  • =LEVENESTEST(B1:C5,.05,0)
LEVENES TEST
DATA-0 DATA-1
Median 2840 2834
Mean 2867.8 2895.6
Variance 16923.7 51713.3
Count 5 5
df 4 4
SUMMARY OUTPUT
LEVENESTEST STATISTICS
W 1.0439235110342522
F-Test 0.38245649772919
a 0.05
F 0.32726010523405
p 1 & 2 Tail 0.1524069466470822 0.3048138932941644

Related Videos

Levene's Test

See Also


References

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