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Box-Cox Transformation

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  • #36968

    Henderson
    Participant

    The data I collected was non-normal.  I performed the Box-Cox transformation with a lambda = -0.225 which yielded normality.  After performing basic staitstics (mean and standard deviation), I wanted to convert those tranformations back to understandable numbers.  However, when I try to transform the numbers back to a ‘real’ number, the results are totally not representative of my data set.  I want to report the mean, standard deviation and 3s range.  My data is below; can someone please help me transform data back?!
    Descriptive Statistics: Transformed WLR
    Variable       N N* Mean SE Mean StDev Minimum Q1 Median
    Transformed  16 0 0.9149 0.0503 0.2014 0.5780 0.7286 0.9529
    Variable         Q3 Maximum
    Transformed  1.0836 1.1690
    WLR                            Transformed WLR11.4                                        0.578054.6                                          0.709148.3                                          0.620882.9                                          0.786795.7                                          0.675710.7                                          1.083640.7                                          1.083640.8                                          1.051541.1                                          0.978762.2                                          0.837302.4                                          0.821051.1                                          0.978760.5                                          1.168950.5                                          1.168951.4                                          0.927020.5                                          1.16895

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    #107768

    Chris Seider
    Participant

    Dear Melissa,
    I think you can find details in Minitab help but just take your mean, s.d., etc to the (-1/0.225) power.  Doesn’t work if lambda is zero, but then you would take the inverse log base 10.
    If you lambda was 0.333, you would take transformed mean and cube it.
    I would caution you reporting the mean, s.g., etc. after you reverse the transformation–they really mean nothing to the casual observer.  You distribution obviously changes so the numbers won’t “look” like your real mean, etc.

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    #107772

    Henderson
    Participant

    Can the s.d also be transformed in the same manner?  If so, then why is the s.d so unrepresentative of my data set.

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    #107774

    roy e.
    Member

    Melissa I would caution you against the used of back-transformed parameters directly.  If you data is not normal on the first place, What would be the interest of describing it with parameters typicaly used for normal distributed populations?
    Having said that, you can calculate the mean and median directly with your data, without transforming.  Be aware that for not normally distributed data the median may be a more meaninful descriptor that the mean.  Regarding SD, I rather suggest to used another descriptor of error, such as variance.
    Box-Cox is useful to transfor the data, get a “normalized”  SD and calculate control limits.  then you can back-transform those control limits to your original data.

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    #107775

    Chris Seider
    Participant

    Melissa,
    Yes, you can apply the same reverse transformation to your standard deviation as your mean.  However, remember that the reversed transformed s.d. does NOT represent the standard deviation of the original data.  That’s why I caution even reporting the reverse transformed data. 
    People use Box Cox transformations to make a better description of their process.  After a change has occurred, hopefully an improvement you made, then they will use the same transform on the new data and see if the transformed mean or standard deviation has changed–using the statistics found with normal distributions.

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    #107816

    DD
    Participant

    Hi
    Once you have decided to transform the data after thoughroughly undrestanding your process characteristics(assuming there is no other way of getting the normal distUse a Lambda of -0.5 as it gives you a better transform into a Normal dist(see below) I have used Minitab14 to get this
    Descriptive Statistics
    N N* Mean StDev Median Minimum Maximum Skewness Kurtosis
    16 0 2.8 3.19186 1.25 0.5 11.4 1.77312 2.65146
    Box-Cox transformation for Normal distribution: Lambda = -0.5
    Goodness of Fit Test
    Distribution AD P LRT P
    Normal 1.575 (<) 0.005
    Normal (After Transformation) 0.376 0.369
    Lognormal 0.440 0.255
    3-Parameter Lognormal 0.816 * 0.001
    Exponential 0.646 0.304
    2-Parameter Exponential 1.169 0.032 0.013
    Weibull 0.633 0.088
    3-Parameter Weibull 0.377 0.432 0.000
    Smallest Extreme Value 1.956 (<) 0.010
    Largest Extreme Value 1.224 (<) 0.010
    Gamma 0.723 0.074
    3-Parameter Gamma 0.260 * 0.000
    Logistic 1.321 (<) 0.005
    Loglogistic 0.449 0.217
    3-Parameter Loglogistic 0.650 * 0.001
    ML Estimates of Distribution Parameters
    Distribution Location Shape Scale Threshold
    Normal* 2.80000 3.19186
    Normal (After Transformation) 0.87116 0.39277
    Lognormal* 0.50351 1.03606
    3-Parameter Lognormal -0.73721 2.48225 0.49500
    Exponential 2.80000
    2-Parameter Exponential 2.30500 0.49500
    Weibull 0.99052 2.78697
    3-Parameter Weibull 0.54634 1.47157 0.49500
    Smallest Extreme Value 4.54748 3.90589
    Largest Extreme Value 1.55563 1.76479
    Gamma 1.08674 2.57651
    3-Parameter Gamma 0.41591 5.54211 0.49500
    Logistic 2.18928 1.55931
    Loglogistic 0.43032 0.60239
    3-Parameter Loglogistic -0.42644 1.40205 0.49500
    * Scale: Adjusted ML estimate
    Here is the transformed Data
    WLR Transformed Data Back Transform11.4 0.29617 11.44.6 0.46625 4.68.3 0.34711 8.32.9 0.58722 2.95.7 0.41885 5.70.7 1.19523 0.70.7 1.19523 0.70.8 1.11803 0.81.1 0.95346 1.12.2 0.67420 2.22.4 0.64550 2.41.1 0.95346 1.10.5 1.41421 0.50.5 1.41421 0.51.4 0.84515 1.40.5 1.41421 0.5
    To Transform it back I have used the calculator functionof minitab using the formula ANTILOG( – 2*LOGT(‘Transformed Data’))
    Hope this helps
    DD

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