It is important to consider the family error rate when making multiple comparisons, because your chances of committing a type I error for a series of comparisons is greater than the error rate for any one comparison alone. If the adjusted p-value is less than alpha, reject the null hypothesis and conclude that the difference between a pair of group means is statistically significant.
The adjusted p-value also represents the smallest family error rate at which a particular null hypothesis is rejected. Adjusted sums of squares are measures of variation for different components of the model. The order of the predictors in the model does not affect the calculation of the adjusted sum of squares. In the Analysis of Variance table, Minitab separates the sums of squares into different components that describe the variation due to different sources.
Minitab uses the adjusted sums of squares to calculate the p-value for a term. Minitab also uses the sums of squares to calculate the R 2 statistic. Usually, you interpret the p-values and the R 2 statistic instead of the sums of squares. A boxplot provides a graphical summary of the distribution of each sample.
The boxplot makes it easy to compare the shape, the central tendency, and the variability of the samples. Use a boxplot to examine the spread of the data and to identify any potential outliers. Boxplots are best when the sample size is greater than Examine the spread of your data to determine whether your data appear to be skewed. When data are skewed, the majority of the data are located on the high or low side of the graph.
Skewed data indicates that the data might not be normally distributed. Often, skewness is easiest to detect with an individual value plot, a histogram, or a boxplot.
The boxplot with right-skewed data shows average wait times. Most of the wait times are relatively short, and only a few of the wait times are longer. The boxplot with left-skewed data shows failure rate data. A few items fail immediately, and many more items fail later. If your data are severely skewed and you have a small sample, consider increasing your sample size.
Outliers, which are data values that are far away from other data values, can strongly affect your results. Often, outliers are easiest to identify on a boxplot. Try to identify the cause of any outliers. Correct any data-entry errors or measurement errors. Consider removing data values for abnormal, one-time events special causes. Then, repeat the analysis. These confidence intervals CI are ranges of values that are likely to contain the true mean of each population. The confidence intervals are calculated using the pooled standard deviation.
Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. But, if you repeat your sample many times, a certain percentage of the resulting confidence intervals contain the unknown population parameter.
The percentage of these confidence intervals that contain the parameter is the confidence level of the interval. Use the confidence interval to assess the estimate of the population mean for each group. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation.
If the interval is too wide to be useful, consider increasing your sample size. In these results, each blend has a confidence interval for its mean hardness. The multiple comparison results for these data show that Blend 4 is significantly harder than Blend 2. That Blend 4 is harder than Blend 2 does not show that Blend 4 is hard enough for the intended use of the paint. The confidence interval for the group mean is better for judging whether Blend 4 is hard enough.
The total degrees of freedom DF are the amount of information in your data. The analysis uses that information to estimate the values of unknown population parameters. The total DF is determined by the number of observations in your sample. The DF for a term show how much information that term uses. Increasing your sample size provides more information about the population, which increases the total DF.
Increasing the number of terms in your model uses more information, which decreases the DF available to estimate the variability of the parameter estimates. If two conditions are met, then Minitab partitions the DF for error.
The first condition is that there must be terms you can fit with the data that are not included in the current model. For example, if you have a continuous predictor with 3 or more distinct values, you can estimate a quadratic term for that predictor. If the model does not include the quadratic term, then a term that the data can fit is not included in the model and this condition is met.
The second condition is that the data contain replicates. Replicates are observations where each predictor has the same value. For example, if you have 3 observations where pressure is 5 and temperature is 25, then those 3 observations are replicates. If the two conditions are met, then the two parts of the DF for error are lack-of-fit and pure error.
The DF for lack-of-fit allow a test of whether the model form is adequate. The lack-of-fit test uses the degrees of freedom for lack-of-fit. The more DF for pure error, the greater the power of the lack-of-fit test. The differences between the sample means of the groups are estimates of the differences between the populations of these groups.
Because each mean difference is based on data from a sample and not from the entire population, you cannot be certain that it equals the population difference. To better understand the differences between population means, use the confidence intervals. Look in the standard deviation StDev column of the one-way ANOVA output to determine whether the standard deviations are approximately equal.
Use the individual confidence intervals to identify statistically significant differences between the group means, to determine likely ranges for the differences, and to determine whether the differences are practically significant. Fisher's individual tests table displays a set of confidence intervals for the difference between pairs of means. The individual confidence level is the percentage of times that a single confidence interval includes the true difference between one pair of group means, if you repeat the study.
Individual confidence intervals are available only for Fisher's method. All of the other comparison methods produce simultaneous confidence intervals.
Controlling the individual confidence level is uncommon because it does not control the simultaneous confidence level, which often increases to unacceptable levels.
If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons. The confidence intervals indicate the following: The confidence interval for the difference between the means of Blend 4 and 2 extends from 4.
This range does not include zero, which indicates that the difference between these means is statistically significant. The confidence interval for the difference between the means of Blend 2 and 1 extends from The confidence interval for the difference between the means of Blend 4 and 3 extends from 0. The confidence intervals for all the remaining pairs of means include zero, which indicates that the differences are not statistically significant.
However, the simultaneous confidence level indicates that you can be only Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the terms and model. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.
If you want to use the F-value to determine whether to reject the null hypothesis, compare the F-value to your critical value. You can calculate the critical value in Minitab or find the critical value from an F-distribution table in most statistics books. Use the grouping information table to quickly determine whether the mean difference between any pair of groups is statistically significant. The grouping column of the Grouping Information table contains columns of letters that group the factor levels.
Groups that do not share a letter have a mean difference that is statistically significant. If the grouping table identifies differences that are statistically significant, use the confidence intervals of the differences to determine whether the differences are practically significant. In these results, the table shows that group A contains Blends 1, 3, and 4, and group B contains Blends 1, 2, and 3.
Blends 1 and 3 are in both groups. Differences between means that share a letter are not statistically significant. Blends 2 and 4 do not share a letter, which indicates that Blend 4 has a significantly higher mean than Blend 2.
The histogram of the residuals shows the distribution of the residuals for all observations. Because the appearance of a histogram depends on the number of intervals used to group the data, don't use a histogram to assess the normality of the residuals. Instead, use a normal probability plot. A histogram is most effective when you have approximately 20 or more data points. In these results, the table shows that group A contains Blends 1, 3, and 4, and group B contains Blends 1, 2, and 3.
Blends 1 and 3 are in both groups. Differences between means that share a letter are not statistically significant. Blends 2 and 4 do not share a letter, which indicates that Blend 4 has a significantly higher mean than Blend 2. In these results, the confidence intervals indicate the following: The confidence interval for the difference between the means of Blend 2 and 4 is 3.
This range does not include zero, which indicates that the difference is statistically significant. The confidence intervals for the remaining pairs of means all include zero, which indicates that the differences are not statistically significant. Each individual confidence interval has a confidence level of This result indicates that you can be S is measured in the units of the response variable and represents the how far the data values fall from the fitted values.
The lower the value of S, the better the model describes the response. However, a low S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions. R 2 is the percentage of variation in the response that is explained by the model. The higher the R 2 value, the better the model fits your data.
A high R 2 value does not indicate that the model meets the model assumptions. Use predicted R 2 to determine how well your model predicts the response for new observations.
Models that have larger predicted R 2 values have better predictive ability. A predicted R 2 that is substantially less than R 2 may indicate that the model is over-fit.
An over-fit model occurs when you add terms for effects that are not important in the population, although they may appear important in the sample data. The model becomes tailored to the sample data and therefore, may not be useful for making predictions about the population. Predicted R 2 can also be more useful than adjusted R 2 for comparing models because it is calculated with observations that are not included in the model calculation.
In these results, the factor explains S indicates that the standard deviation between the data points and the fitted values is approximately 3. Use the residual plots to help you determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and you should use caution when you interpret the results. The time to complete the marathon is the outcome dependent variable.
This study design is illustrated schematically in the diagram below:. When you might use this test is continued on the next page. What does this test do? In contrast, the mean and mode can vary in skewed distributions. Because the range formula subtracts the lowest number from the highest number, the range is always zero or a positive number.
In statistics, the range is the spread of your data from the lowest to the highest value in the distribution. It is the simplest measure of variability. While central tendency tells you where most of your data points lie, variability summarizes how far apart your points from each other. Data sets can have the same central tendency but different levels of variability or vice versa.
Together, they give you a complete picture of your data. Variability is most commonly measured with the following descriptive statistics :. Variability tells you how far apart points lie from each other and from the center of a distribution or a data set.
While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero. For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy. A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test.
It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data i. The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution a. In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data. A t -score a.
The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution. The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown. The t -distribution forms a bell curve when plotted on a graph.
It can be described mathematically using the mean and the standard deviation. The correlation coefficient only tells you how closely your data fit on a line, so two datasets with the same correlation coefficient can have very different slopes. Correlation coefficients always range between -1 and 1.
The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.
The absolute value of a number is equal to the number without its sign. The absolute value of a correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation. A correlation coefficient is a single number that describes the strength and direction of the relationship between your variables.
Different types of correlation coefficients might be appropriate for your data based on their levels of measurement and distributions. A power analysis is a calculation that helps you determine a minimum sample size for your study. If you know or have estimates for any three of these, you can calculate the fourth component. In statistical hypothesis testing , the null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
Statistical analysis is the main method for analyzing quantitative research data. It uses probabilities and models to test predictions about a population from sample data.
The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one. To indirectly reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power. The risk of making a Type I error is the significance level or alpha that you choose. The significance level is usually set at 0.
In statistics, ordinal and nominal variables are both considered categorical variables. Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them.
In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative a Type II error. Your study might not have the ability to answer your research question. While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.
Statistical significance is denoted by p -values whereas practical significance is represented by effect sizes. There are dozens of measures of effect sizes. Effect size tells you how meaningful the relationship between variables or the difference between groups is.
A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications. Using descriptive and inferential statistics , you can make two types of estimates about the population : point estimates and interval estimates. Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.
Standard error and standard deviation are both measures of variability. The standard deviation reflects variability within a sample, while the standard error estimates the variability across samples of a population. The standard error of the mean , or simply standard error , indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.
To figure out whether a given number is a parameter or a statistic , ask yourself the following:. If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.
If the answer is no to either of the questions, then the number is more likely to be a statistic. The arithmetic mean is the most commonly used mean. But there are some other types of means you can calculate depending on your research purposes:. You can find the mean , or average, of a data set in two simple steps:.
This method is the same whether you are dealing with sample or population data or positive or negative numbers. Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.
The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset. Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.
In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data. The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision. AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting.
In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable. You can test a model using a statistical test. The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters K used to reach that likelihood. The AIC function is 2K — 2 log-likelihood. Lower AIC values indicate a better-fit model, and a model with a delta-AIC the difference between the two AIC values being compared of more than -2 is considered significantly better than the model it is being compared to.
The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe.
It penalizes models which use more independent variables parameters as a way to avoid over-fitting. AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data. If you are only testing for a difference between two groups, use a t-test instead. The formula for the test statistic depends on the statistical test being used. Generally, the test statistic is calculated as the pattern in your data i.
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