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Excel statistical functions: GROWTH
Article ID: 828526 - View products that this article applies to.
This article describes the GROWTH function in Microsoft Office Excel 2003 and in later versions of Excel, illustrates how the function is used, and compares results of the function for Excel 2003 and for later versions of Excel with results of GROWTH in earlier versions of Excel. GROWTH is evaluated by calling the related function, LINEST. Extensive changes to LINEST for Excel 2003 and for later versions of Excel are summarized, and their implications for GROWTH are noted.
Microsoft Excel 2004 for Macintosh informationThe statistical functions in Excel 2004 for Mac were updated by using the same algorithms that were used to update the statistical functions in Excel 2003 and in later versions of Excel. Any information in this article that describes how a function works or how a function was modified for Excel 2003 or for later versions of Excel also applies to Excel 2004 for Mac.
The GROWTH(known_y's, known_x's, new_x's, constant) function is used to perform a regression analysis where an exponential curve is fitted. A least squares criterion is used, and GROWTH tries to find the best fit under that criterion. Known_y's represent data on the "dependent variable" and known_x's represent data on one or more "independent variables". The GROWTH Help file discusses rare cases where the second or third argument may be omitted.
Assuming that there are p predictor variables, GROWTH essentially calls LOGEST. LOGEST fits an equation of the form:
Values of the coefficients, b, m1, m2, ..., mp are determined that give the best fit to the y data.
If the last argument "constant" is set to TRUE, you want the regression model to include the multiplicative coefficient b in the regression model. If set to FALSE, b is excluded by essentially setting it to 1. The last argument is optional; if the argument is omitted it is interpreted as TRUE.
For ease of exposition in the remainder of this article, assume that data is arranged in columns so that known_y's is a column of y data and known_x's is one or more columns of x data. Of course the dimensions (lengths) of each of these columns must be equal. New_x's will also be assumed to be arranged in columns and there must be the same number of columns for new_x's as for known_x's. All our observations below are equally true if the data is not arranged in columns, but it is just easier to discuss this single (most frequently used) case.
After you compute the best fit regression model (by essentially calling Excel's LOGEST function), GROWTH returns predicted values that are associated with new_x's.
This article uses examples to show how GROWTH relates to LOGEST and to point out problems with LOGEST in versions of Excel that are earlier than Excel 2003 that translate to problems with GROWTH. GROWTH effectively calls LOGEST, executes LOGEST, uses regression coefficients in LOGEST output in its calculation of predicted y values that are associated with each row of new_x's, and presents this column of predicted y values to you. Therefore, you must know about problems in the execution of LOGEST. When LOGEST is called, it in turn effectively calls LINEST. While code for GROWTH and LOGEST have not been rewritten for Excel 2003 and for later versions of Excel, extensive changes (and improvements) in LINEST code have been made.
As supplements to this article, the following article about LINEST is highly recommended. It contains several examples and documents problems with LINEST in versions of Excel that are earlier than Excel 2003.
For more information about LINEST, click the following article number to view the article in the Microsoft Knowledge Base:
828533The LINEST Help file, as revised for Excel 2003, is also recommended.
(http://support.microsoft.com/kb/828533/ )Description of the LINEST function in Excel 2003 and in Excel 2004 for Mac
The following article about LOGEST explains how LOGEST interacts with LINEST. These details are omitted here.
For more information, click the following article number to view the article in the Microsoft Knowledge Base:
828528Because the focus in this article is on numeric problems in versions of Excel that are earlier than Excel 2003, this article does not have many practical examples of the use of GROWTH. The Help file in GROWTH contains useful examples.
(http://support.microsoft.com/kb/828528/ )Excel statistical functions: LOGEST
The arguments, known_y's, known_x's, and new_x's must be arrays or cell ranges that have related dimensions. If known_y's is one column by m rows then known_x's is c columns by m rows where c is greater than or equal to one. C is the number of predictor variables; m is the number of data points. New_x's must then be c columns by r rows where r is greater than or equal to one. (Similar relationships in dimensions must hold if data is laid out in rows instead of columns.) Constant is a logical argument that must be set to TRUE or FALSE (or 0 or 1 that Excel interprets as FALSE or TRUE, respectively). The last three arguments to GROWTH are all optional; see the GROWTH Help file for options of omitting the second argument, third argument, or both; omitting the fourth argument is interpreted as TRUE.
The most common usage of GROWTH includes two ranges of cells that contain the data, such as GROWTH(A1:A100, B1:F100, B101:F108, TRUE). Note that because there is typically more than one predictor variable, the second argument in this example contains multiple columns. In this example, there are one hundred subjects, one dependent variable value (known_y) for each subject, and five dependent variable values (known_x's) for each subject. There are eight additional hypothetical subjects where you want to use GROWTH to compute predicted y values.
Example of usageAn Excel worksheet example is provided to illustrate the following key concepts:
To illustrate the GROWTH function, create a blank Excel worksheet, copy the following table, select cell A1 in your blank Excel worksheet and then paste the entries so that the table following fills cells A1:K35 in your worksheet.
Note After you paste this table in your new Excel worksheet, click the Paste Options button, and then click Match Destination Formatting. With the pasted range still selected, use one of the following procedures, as appropriate for the version of Excel that you are running:
Collapse this tableExpand this table
GROWTH and LOGEST can be viewed as interacting in the following steps:
Predictor columns (known_x's) are collinear if at least one column, c, can be expressed as a sum of multiples of others, c1, c2, and other columns. Column c is frequently called redundant because the information that it contains can be constructed from the columns c1, c2, and other columns. The fundamental principle in the existence of collinearity is that results should be unaffected by whether a redundant column is included in the original data or removed from the original data. Because LINEST in versions of Excel that are earlier than Excel 2003 did not look for collinearity, this principle was easily violated. Predictor columns are almost collinear if at least one column, c, can be expressed as almost equal to a sum of multiples of others, c1, c2, and other columns. In this case "almost equal" means a very small sum of squared deviations of entries in c from corresponding entries in the weighted sum of c1, c2, and other columns. "Very small" might be less than 10^(-12), for example.
The first model, in rows 10 to 12, uses columns B and C as predictors and requests Excel to model the constant (last argument set to TRUE). Excel then effectively inserts an additional predictor column that looks just like cells D2:D6. It is easy to notice that entries in column C in rows 2 to 6 are exactly equal to the sum of corresponding entries in columns B and D. Therefore, there is collinearity present because column C is a sum of multiples of the following items:
The second model, in rows 14 to 16, is one that any version of Excel can handle successfully. There is no collinearity, and the user again requests Excel to model the constant. This model is included here for the following reasons:
In the second model in rows 30 to 35, there is no collinearity and no column removed. You can see that the predicted y values are the same in both models. This issue occurs because removing a redundant column that is a sum of multiples of others does not reduce the goodness of fit of the resulting model. Such columns are removed precisely because they represent no value added in trying to find the best least squares fit. Also, if you examine the LOGEST output in cells I23:K35 in Excel 2003 and in later versions of Excel, you will notice that the last three rows of the output tables are the same. Additionally, the entries in cells I31:J32 and cells J24:K25 coincide. This demonstrates that the same results are obtained when column C is included in the model, but found to be redundant (output in cells I24:K28) as when column C was eliminated before LOGEST was run (output in cells I31:J35). This satisfies the fundamental principle in the existence of collinearity.
In cells A18:C21, Microsoft uses data from Excel 2003 and from later versions of Excel to illustrate how GROWTH takes LOGEST output and computes the relevant predicted y-values. By examining the formulas in cells A20:A21 and cells C20:C21, you can see how LOGEST coefficients are combined with new_x's data in cells B7:C8 for each of the two models (using columns B, C as predictors; using only column B as a predictor).
Collinearity is identified in LOGEST in Excel 2003 and in later versions of Excel because LOGEST calls LINEST. LINEST uses a completely different approach to solving for the regression coefficients. This approach is QR Decomposition. The LINEST article contains a walkthrough of the QR Decomposition algorithm for a small example.
Summary of results in earlier versions of ExcelGROWTH results are adversely affected in versions of Excel that are earlier than Excel 2003 because of inaccurate results in LOGEST that, in turn, stem from inaccurate results in LINEST.
LINEST was calculated using an approach that paid no attention to collinearity issues. The existence of collinearity caused roundoff errors, inappropriate standard errors of regression coefficients, and inappropriate degrees of freedom. Sometimes roundoff problems are sufficiently severe that LINEST filled its output table with #NUM!. If, as in the great majority of cases in practice, you can be confident that there were not collinear (or almost collinear) predictor columns, then LINEST would generally provide acceptable results. Therefore, users of GROWTH can be similarly reassured if they can see the absence of collinear (or almost collinear) predictor columns.
Summary of results in Excel 2003 and in later versions of ExcelImprovements in LINEST include switching to the QR Decomposition method of determining regression coefficients. QR Decomposition has the following advantages:
ConclusionsGROWTH's performance has been improved because LINEST has been greatly improved for Excel 2003 and for later versions of Excel. Improvements in LINEST also affect LOGEST, because LOGEST is essentially called by GROWTH. Users of earlier versions of Excel should verify that predictor columns are not collinear before they use GROWTH.
Much of the material presented in this article and in the LINEST article might at first appear alarming to users of versions of Excel that are earlier than Excel 2003. However, it should be noted that collinearity is a problem in only a small percentage of cases. Earlier versions of Excel give acceptable GROWTH results when there is no collinearity.
Fortunately, improvements in LINEST also affect the Analysis ToolPak's linear regression tool (this tool calls LINEST) and two other related Excel functions: LOGEST and TREND.