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Interpreting interaction effects
互動效應的詮釋

This web page contains various Excel templates which help interpret two-way and three-way interaction effects.
本網頁包含各種 Excel 範本,有助於詮釋雙向和 三方互動效應。

They use procedures by Aiken and West (1991), Dawson (2014) and Dawson and Richter (2006) to plot the interaction effects, and in the case of three way interactions test for significant differences between the slopes.
他們使用程序 作者:Aiken and West (1991)、Dawson (2014) 以及 Dawson 和 Richter (2006) 繪製的 互動效應,以及在 三向互動測試是否顯著 之間的差異 斜坡。

You can either use the Excel templates directly from this page, or download them to your computer by right-clicking on the relevant links.
您可以使用 Excel 範本 直接從本頁面,或 用滑鼠右鍵按一下 相關連結。

Two-way interactions 雙向互動

To test for two-way interactions (often thought of as a relationship between an independent variable (IV) and dependent variable (DV), moderated by a third variable), first run a regression analysis, including both independent variables (referred to hence as the IV and moderator) and their interaction (product) term.
為了測試雙向互動(通常 之間的關係 自變量 (IV) 與因變量 (DV),由第三位調節 變數),首先執行回歸分析、 包括兩個獨立變數 (因此稱為 IV 和主持人)和 其互動 (乘積) 項。

It is recommended that the independent variable and moderator are centred before calculation of the product term, although this is not essential. The product term should be significant in the regression equation in order for the interaction to be interpretable.
建議在計算乘積項之前,先將自變數和主持人居中、 雖然這並非必要條件。工作地點 乘積項在迴歸中應該是顯著的 方程式,以便 互動才能解釋。

You can then plot the interaction effect using the following Excel template. You will need to enter the unstandardised regression coefficients (including intercept/constant) and means & standard deviations of the IV and moderator in the cells indicated.
然後,您可以使用下列 Excel 模版繪製交互作用效果圖。您需要在指定的單元格中輸入未標準化的回歸係數(包括截距/常量)以及 IV 和調節因子的平均值和標準差。

If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you centre (or standardise) all control variables first (although even if you don’t the pattern, and therefore the interpretation, will be correct).
如果您在迴歸中有控制變數,除非您先將所有控制變數居中(或標準化),否則因變數在繪圖上顯示的值會不準確(雖然即使您沒有居中,模式以及詮釋也會正確)。

If your moderator is binary, ensure you enter the actual values of this variable in cells B31 and B32.
2-way_linear_interactions.xls
2-way_linear_interactions.xls 如果您的主持人是二進制,請確保您在單元格 B31 和 B32 中輸入此變數的實際值。

Simple slope tests. This template also allows you to perform simple slope tests – these are conditional hypothesis tests of whether the relationship between IV and DV is significant at a particular value of the moderator.
簡單斜率測試。這個範本也允許您執行簡單的斜率檢驗,這些是條件假設檢驗,檢驗 IV 和 DV 之間的關係在特定的主持人值時是否顯著。

If your moderator is a numerical variable, these are not necessary, but may be useful at specific, theoretically interesting values of the moderator.
如果您的控制變數是數值變數,這些不是必要的,但在特定的、理論上有趣的控制變數值時可能會有用。

Therefore where possible, meaningful values should be chosen, rather than just one standard deviation above and below the mean (which is where they will be tested by default if you leave cells B31 & B32 blank); otherwise the tests are arbitrary in nature and quite possibly meaningless.
因此,在可能的情況下,應選擇有意義的值,而不是僅選擇高於或低於平均值的一個標準差(如果您將單元 B31 和 B32 留空,則會在預設情況下進行測試);否則,測試在本質上是任意的,而且很可能毫無意義。

To run simple slope tests, you will also need to request the coefficient covariance matrix as part of the regression output. If you are using SPSS, this can be done by selecting "Covariance matrix" in the "Regression Coefficients" section of the "Statistics" dialog box.
若要執行簡單的斜率測試,您還需要要求將係數共變方差矩陣作為迴歸輸出的一部分。如果您使用的是 SPSS,可以在「統計」對話方塊的「迴歸係數」區段中選擇「共變異矩陣」。

Note that the variance of a coefficient is the covariance of that coefficient with itself - i.e. can be found on the diagonal of the coefficient covariance matrix.
請注意,係數的方差是該係數與其本身的共變方差 - 即可以在係數共變方差矩陣的對角線上找到。

For non-linear two-way interactions (including generalised linear models), you might want to use one of the following templates:
對於非線性雙向交互作用 (包括廣義線性模型),您可能需要使用下列範本之一:

A note about centred and standardised variables. Centred variables are the same as the original version, with the variable mean subtracted so that the new mean is zero.
關於居中變數和標準化變數的說明。居中變數與原始版本相同,只是減去變數的平均值,使新的平均值為零。

Standardised variables are those that are both centred around zero and are scaled so that they have a standard deviation of 1. Following Aiken and West (1991), I recommend that for analysis variables are centred.
標準化變數是指 都是以零為中心,且比例為 跟隨 Aiken 和 West (1991),我建議在分析時 變數居中。

Some researchers may prefer to use standardised variables. Each gives some advantages in interpreting the coefficients - see Dawson (2014) for more about this (reference below).
有些研究人員可能偏好使用標準化變數。每種變量都提供了一些 解釋係數的優勢 - 請參閱 Dawson (2014) 以獲得更多相關資訊(參考資料如下)。

However, the results obtained should be identical whichever method you use.
然而 所得結果應完全相同。 您使用的方法。

If you choose to analyse centred (or standardised) variables, you should use the regular versions of the Excel templates, and enter the mean of the variables as zero (and standard deviation as 1 if using standardised versions).
如果您選擇以分析為中心 (或標準化)變數,您應該使用 Excel 範本的一般版本、 並輸入變數的平均值為零(如果使用標準化版本,則標準差為 1)。

And for those readers who use the US version of English, for "centred" and "standardised", read "centered" and "standardized"!
對於使用美國版英語的讀者,"centred 「和 」standardised 「應為 」centered 「和 」standardized"!


Three-way interactions 三方互動

To test for three-way interactions (often thought of as a relationship between a variable X and dependent variable Y, moderated by variables Z and W), run a regression analysis, including all three independent variables, all three pairs of two-way interaction terms, and the three-way interaction term.
為了測試三方互動(通常 之間的關係 變數 X 與因變數 Y,由 變數 Z 和 W),執行回歸分析、 包括 所有三個自變量、所有三對 雙向互動項、 和三方互動項。

It is recommended that all the independent variable are centred (or standardised) before calculation of the product terms, although this is not essential. As with two-way interactions, the interaction terms themselves should not be centred (or standardised) after calculation.
建議在計算乘積項目之前,先將所有自變數居中(或標準化)、 雖然這並非必要條件。如 與雙向互動,互動項 自己不應 計算後居中(或標準化)。

The three-way interaction term should be significant in the regression equation in order for the interaction to be interpretable.
三方 互動項應為 在迴歸方程式中顯著,以便 的互動 可解釋。

If you wish to use the Dawson & Richter (2006) test for differences between slopes, you should request the coefficient covariance matrix as part of the regression output.
如果您希望使用 Dawson & Richter (2006) 測試差異 斜率之間,您應該要求系數 作為 回歸輸出。

If you are using SPSS, this can be done by selecting "Covariance matrix" in the "Regression Coefficients" section of the "Statistics" dialog box.
如果您使用 SPSS,此 可透過選擇 「回归系数 」中的 「协方差矩阵」 區段 的「統計」對話方塊。

Note that the variance of a coefficient is the covariance of that coefficient with itself - i.e. can be found on the diagonal of the coefficient covariance matrix.
請注意 係數的方差就是該係數的協方差。 與其本身的系數 - 即可以在 係數共變數矩陣的對角線。

You can then plot the interaction effect using the following Excel template. You will need to enter the unstandardised regression coefficients (including intercept/constant) and means & standard deviations of the three independent variables (X, Z and W) in the cells indicated.
然後,您可以使用下列 Excel 模版來繪製交互作用效果圖。您需要在指定的單元格中輸入三個自變量 (X、Z 和 W) 的非標準化迴歸係數(包括截距/常數)、平均值和標準差。

If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you centre (or standardise) all control variables first (although even if you don’t the pattern, and therefore the interpretation, will be correct).
如果您在迴歸中有控制變數,除非您先將所有控制變數居中(或標準化),否則因變數在繪圖上顯示的值會不準確(雖然即使您沒有居中,模式以及詮釋也會正確)。

To use the test of slope differences, you should also enter the covariances of the XZ, XW and XZW coefficients from the coefficient covariance matrix, and the total number of cases and number of control variables in your regression.
要使用斜率差測試,您還應該輸入系數共變異矩陣中 XZ、XW 和 XZW 系數的共變異,以及迴歸中的個案總數和控制變數數。

If any of your variables is binary, ensure you enter the actual values of this variable where stated.
3-way_linear_interactions.xls
如果您的任何變數是二進位變數,請確保在註明處輸入該變數的實際值。3-way_linear_interactions.xls

Simple slope tests. As with the 2-way interactions above, this template also allows you to perform simple slope tests, as well as the slope difference tests. See my warnings above about the use of simple slope tests, however.
簡單斜率測試。與上面的 2 向交互作用一樣,此模板也允許您執行簡單斜率測試以及斜率差異測試。但是,請參閱我在上面關於使用簡單斜率測試的警告。

For non-linear three-way interactions (including generalised linear models), you might want to use one of the following templates:
對於非線性三向交互作用(包括廣義線性模型),您可能需要使用下列範本之一:


Quadratic Effects 二次方效應

If you wish to plot a quadratic (curvilinear) effect, you can use one of the following Excel templates.
如果您想繪製二次方 (曲線)效果,您可以使用其中一個 下列 Excel 範本。

In each case, you test the quadratic effect by including the main effect (the IV) along with its squared term (i.e. the IV*IV) in the regression.
在每種情況下,您都要測試 透過包含主效應的二次方效應 (IV)及其平方項(即 IV*IV)的迴歸。

In the case of a simple (unmoderated) relationship, the significance of the squared term determines whether there is a quadratic effect.
在簡單 (未經調變)的關係,其意義 平方項決定是否存在二次方 效果。

If you are testing a moderated quadratic relationship, it is the significance of the interaction between the squared term and the moderator(s) that determines whether there is a moderated effect.
如果您要測試修正的二次方 關係,它的意義在於 平方項與 主持人決定是否有一個 緩和效應。

Note that despite this, all lower order terms need to be included in the regression: so, if you have an independent variable A and moderators B and C, then to test whether there is a three-way interaction you would need to enter all the following terms: A, A*A, B, C, A*B, A*C, A*A*B, A*A*C, B*C, A*B*C, A*A*B*C.
請注意,儘管如此,所有較低 階次項目需要包含在迴歸中: 因此,如果您有一個自變數 A,並且 調節器 B 和 C,然後測試是否存在 三向互動,您需要輸入所有 下列詞彙:a、a*a、b、c、a*b、a*c、a*a*b、 a*a*c、b*c、a*b*c、a*a*b*c。

It is only the last, however, that determines the significance of the three-way quadratic interaction.
這只是最後一次、 然而,這決定了 三向四元交互作用。


Troubleshooting 疑難排解

There are a number of common problems encountered when trying to plot these effects. If you are having problems, consider the following:
嘗試繪製這些效果時會遇到一些常見問題。如果您遇到問題,請考慮以下幾點:

  • If the graph does not appear, it may be because it is off the scale. You can change the scale of the dependent variable by right-clicking on the axis and choosing "Format Axis"
    如果圖形沒有出現,可能是因為圖形偏離了比例。您可以在軸上按一下滑鼠右鍵,然後選擇「格式化軸」來變更因變數的比例。

  • Make sure you enter the unstandardised regression coefficients, whether or not you are using centred/standardised variables
    確保輸入 非標準化 回歸係數,無論您是否使用居中/標準化變數。

  • If you use centred (or standardised) variables, ensure that you calculate the interaction (product) terms from the centred (or standardised) variables, but do not centre (or standardise) the interaction terms themselves
    如果您使用居中(或標準化)變數,請確保根據居中(或標準化)變數計算互動(乘積)項,但不要居中(或標準化)互動項本身。

  • When performing simple slopes or slope difference tests, it is easy to enter the wrong figures for variances & covariances of coefficients!
    執行簡單的 斜率或斜率差測試,很容易 輸入錯誤的差異數字 & 係數的共變數!

    SPSS is prone to printing the covariances in a different order from the regression coefficients themselves, which can be confusing.
    SPSS 容易 以不同的順序列印共變數 來自回歸係數本身、 這可能會造成混淆。

    Also, SPSS automatically prints a correlation matrix of the coefficients above the variance-covariance matrix: ensure that you do not enter these in error.
    此外,SPSS 會自動 列印係數的關聯矩陣 上方的方差-方差矩陣:確保 請勿錯誤輸入。

    Note that the variances of the coefficients are along the diagonal of this matrix: e.g. the variance of the Var1*Var2 coefficient is the covariance of this coefficient with itself.
    請注意 係數的方差沿著 此矩陣的對角線:如 的協方差。 此系數與其本身。

 

If you think there are any errors in these sheets, please contact me, Jeremy Dawson
Jeremy Dawson
.

 

References 參考資料

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, London, Sage.
Aiken, L. S., & West, S. G. (1991). 多重迴歸: Newbury Park, London, Sage.

Dawson, J. F. (2014). Moderation in management research: What, why, when and how. Journal of Business and Psychology, 29, 1-19.
Dawson, J. F. (2014)。管理研究中的調節:什麼、為什麼、何時及如何。Journal of Business and Psychology, 29, 1-19.

(This article includes information about most of the tests included on this page, as well as much more! Click here for this article.)
(這篇文章包含本頁面中大部分測試的相關資訊,以及更多資訊!點選此處查看本文。)

Dawson, J. F., & Richter, A. W. (2006). Probing three-way interactions in moderated multiple regression: Development and application of a slope difference test. Journal of Applied Psychology, 91, 917-926.
Dawson, J. F., & Richter, A. W. (2006)。調節多元回归中的三方交互作用探究:斜率差異測試的開發與應用。Journal of Applied Psychology, 91, 917-926.

 

Other online resources 其他線上資源

Kristopher Preacher's web site contains templates for testing simple slopes, and findings regions of significance, for both 2-way and 3-way interactions.
Kristopher Preacher 的網站包含測試簡單斜率和發現顯著性區域的範本,適用於 2 向和 3 向交互作用。

It also includes options for hierarchical linear modelling (HLM) and latent curve analysis.
它還包括 分層線性模型 (HLM) 的選項 和潛在曲線分析。

Yung-jui Yang's web site contains SAS macros to plot interaction effects and run the slope difference tests for three-way interactions
Yung-jui Yang's web site 包含 SAS 巨集來繪製交互作用效應和執行三方交互作用的斜率差測試

Cameron Brick's web site contains instructions on how to plot a three-way interaction and test for differences between slopes in Stata
包含如何在 Stata 中繪製三方交互作用圖和測試斜率之間差異的說明

 

 

Legacy versions of Excel templates
Excel 範本的舊版本

Previously, some different versions of the linear interaction template were available on this page. I strongly recommend using the above versions, but if you want to see one of the older (legacy) versions, you can click on the appropriate file: 2-way_unstandardised.xls 2-way_standardised.xls 2-way_with_binary_moderator.xls 2-way_with_all_options.xls 3-way_unstandardised.xls 3-way_standardised.xls 3-way_with_all_options.xls
在此之前,本頁面上有一些不同版本的線性互動範本。我強烈建議您使用上述版本,但如果您想看較舊的(傳統)版本,可以點選適當的檔案:2-way_unstandardised.xls2-way_standardised.xls2-way_with_binary_moderator.xls2-way_with_all_options.xls3-way_unstandardised.xls3-way_standardised.xls3-way_with_all_options.xls