Interpreting Log Transformations in a Linear Model Log transformations are often recommended for skewed data, such as monetary measures or certain biological and demographic measures. Log transforming data usually has the effect of spreading out clumps of data and bringing together spread-out data.
Estimation of model parameters Once the data is log-transformed, many statistical methods, including linear regression, can be applied to model the resulting transformed data.
Log transforming data usually has the effect of spreading out clumps of data and bringing together spread-out data. For example, below is a histogram of the areas of all 50 US states.
Despite the common belief that the log transformation can decrease the variability of data and make data conform more closely to the normal distribution, this is usually not the case.
I made normal log, log 10, box-cox to transform these data but they are still not normally distributed. I have 6 treatments - I am talking about normal distribution for each single treatment if we ...
$\begingroup$ Ah, but you know much more than that, because after using logs in regression, you know that the results are interpreted differently and you know to take care in back-transforming fitted values and confidence intervals. I'm suggesting that you might not be confused and that you probably already know many of the answers to these four questions, even though you weren't initially ...
Another popular use of the log transformation is to reduce the variability of data, especially in data sets that include outlying observations. Again, contrary to this popular belief, log transformation can often increase – not reduce – the variability of data whether or not there are outliers.
The log-transformation is widely used in biomedical and psychosocial research to deal with skewed data. This paper highlights serious problems in this classic approach for dealing with skewed data. Despite the common belief that the log transformation can decrease the variability of data and make data conform more closely to the normal distribution, this is usually not the case. Moreover, the results of standard statistical tests performed on log-transformed data are often not relevant for the original, non-transformed data.We demonstrate these problems by presenting examples that use simulated data. We conclude that if used at all, data transformations must be applied very cautiously. We recommend that in most circumstances researchers abandon these traditional methods of dealing with skewed data and, instead, use newer analytic methods that are not dependent on the distribution the data, such as generalized estimating equations (GEE).
Thus, if we apply the two-sample t-test to the transformed data, the null hypothesis of the equality of the means becomes, H0:μ1=μ2.
If the data from both samples follow a log-normal distribution, with log-normal (μ1, σ12) for the first sample and (μ2, σ22) for the second sample, then the first sample has the mean exp(μ1+σ12/2) and the second has the mean exp(μ2+σ22/2).If we apply the two-sample t-test to the original data, we are testing the null hypothesis that these two means are equal, H0: exp(μ1+σ12/2)=exp(μ2+σ22/2)
A more fundamental problem is that there is little value in comparing the variability of original versus log-transformed data because they are on totally different scales. In theory we can always find a transformation for any data to make the variability of the transformed version either smaller or larger than that of the original data. For example, if the standard deviation of variable x is σ, then the standard deviation of the scale transformation x/K (K>0) is σ/K; thus by selecting a sufficiently large or small K we can change the standard deviation of the transformed variable x/K to any desired level.
In this case, the log-transformation does remove or reduce skewness. Unfortunately, data arising from many studies do not approximate the log-normal distribution so applying this transformation does not reduce the skewness of the distribution. In fact, in some cases applying the transformation can make the distribution more skewed than the original data.
The two null hypotheses are clearly not equivalent. Although the null hypothesis based on the log-transformed data does test the equality of the means of the two log-transformed samples, the null hypothesis based on the original data does not, since the mean of the original data also involves the parameters, σ12and σ22.Thus, even if no difference is found between the two means of the log-transformed data, it does not mean that there is no differences between the means in the original data of the two samples. For example, if the null hypothesis for the log-transformed data, H0:μ1=μ2, is not rejected for the log-transformed data, it does not imply that the null hypothesis for comparing the means of the original data of the samples, H0: exp(μ1+σ12/2)=exp(μ2+σ22/2), is true, unless the variances of the two samples are the same.
If the absolute value of the t-value is greater than the critical value, you reject the null hypothesis. If the absolute value of the t-value is less than the critical value, you fail to reject the null hypothesis.
Confidence intervals for predicted Y are less precise when the residuals are very small
An estimated regression for a random sample of observations on an assembly line is Defects = 4.4 + 0.055 Speed, where Defects is the number of defects per million parts and Speed is the number of units produced per hour. The estimated standard error is se = 1.11. Suppose that 125 units per hour are produced and the actual (observed) defect rate is Defects = 4.3.
The correlation coefficient r measures the strength of the linear relationship between two variables
A regression with 60 observations and 5 predictors does not violate Evans' Rule.
The independent variables and the dependent variable have a linear relationship
the quadratic model has one additional parameter to estimate.
Yes, since the p-value is less than 0.05.
it can be tested for significance using a t-test
One reason is to make data more “normal”, or symmetric. If we’re performing a statistical analysis that assumes normality , a log transformation might help us meet this assumption. Another reason is to help meet the assumption of constant variance in the context of linear modeling.
Interpret the coefficient as the percent increase in the dependent variable for every 1% increase in the independent variable. Example: the coefficient is 0.198. For every 1% increase in the independent variable, our dependent variable increases by about 0.20%. For x percent increase, calculate 1.x to the power of the coefficient, subtract 1, and multiply by 100. Example: For every 20% increase in the independent variable, our dependent variable increases by about (1.20 0.198 – 1) * 100 = 3.7 percent.
Notice the standardized residuals are trending upward. This is a sign that the constant variance assumption has been violated. Compare this plot to the same plot for the correct model.
Hence the interpretation that a 1% increase in x increases the dependent variable by the coefficient/100.
Only independent/predictor variable (s) is log-transformed. Divide the coefficient by 100. This tells us that a 1% increase in the independent variable increases (or decreases) the dependent variable by (coefficient/100) units. Example: the coefficient is 0.198. 0.198/100 = 0.00198. For every 1% increase in the independent variable, our dependent variable increases by about 0.002. For x percent increase, multiply the coefficient by log (1.x). Example: For every 10% increase in the independent variable, our dependent variable increases by about 0.198 * log (1.10) = 0.02.
If we fit the correct model to the data, notice we do a pretty good job of recovering the true parameter values that we used to generate the data.
Log transforming data usually has the effect of spreading out clumps of data and bringing together spread-out data. For example, below is a histogram of the areas of all 50 US states. It is skewed to the right due to Alaska, California, Texas and a few others. hist (state.area)
Another popular use of the log transformation is to reduce the variability of data, especially in data sets that include outlying observations. Again, contrary to this popular belief, log transformation can often increase – not reduce – the variability of data whether or not there are outliers.
The log-transformation is widely used in biomedical and psychosocial research to deal with skewed data. This paper highlights serious problems in this classic approach for dealing with skewed data. Despite the common belief that the log transformation can decrease the variability of data and make data conform more closely to the normal distribution, this is usually not the case. Moreover, the results of standard statistical tests performed on log-transformed data are often not relevant for the original, non-transformed data.We demonstrate these problems by presenting examples that use simulated data. We conclude that if used at all, data transformations must be applied very cautiously. We recommend that in most circumstances researchers abandon these traditional methods of dealing with skewed data and, instead, use newer analytic methods that are not dependent on the distribution the data, such as generalized estimating equations (GEE).
Thus, if we apply the two-sample t-test to the transformed data, the null hypothesis of the equality of the means becomes, H0:μ1=μ2.
If the data from both samples follow a log-normal distribution, with log-normal (μ1, σ12) for the first sample and (μ2, σ22) for the second sample, then the first sample has the mean exp(μ1+σ12/2) and the second has the mean exp(μ2+σ22/2).If we apply the two-sample t-test to the original data, we are testing the null hypothesis that these two means are equal, H0: exp(μ1+σ12/2)=exp(μ2+σ22/2)
A more fundamental problem is that there is little value in comparing the variability of original versus log-transformed data because they are on totally different scales. In theory we can always find a transformation for any data to make the variability of the transformed version either smaller or larger than that of the original data. For example, if the standard deviation of variable x is σ, then the standard deviation of the scale transformation x/K (K>0) is σ/K; thus by selecting a sufficiently large or small K we can change the standard deviation of the transformed variable x/K to any desired level.
In this case, the log-transformation does remove or reduce skewness. Unfortunately, data arising from many studies do not approximate the log-normal distribution so applying this transformation does not reduce the skewness of the distribution. In fact, in some cases applying the transformation can make the distribution more skewed than the original data.
The two null hypotheses are clearly not equivalent. Although the null hypothesis based on the log-transformed data does test the equality of the means of the two log-transformed samples, the null hypothesis based on the original data does not, since the mean of the original data also involves the parameters, σ12and σ22.Thus, even if no difference is found between the two means of the log-transformed data, it does not mean that there is no differences between the means in the original data of the two samples. For example, if the null hypothesis for the log-transformed data, H0:μ1=μ2, is not rejected for the log-transformed data, it does not imply that the null hypothesis for comparing the means of the original data of the samples, H0: exp(μ1+σ12/2)=exp(μ2+σ22/2), is true, unless the variances of the two samples are the same.