第二十一篇 Likert Scales and Data Analyses
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对下面感兴趣的朋友，请留下你的预计完成时间和邮箱，并发送信息给小编D，谢谢~
本文翻译者：xy_persist 校稿者： liphking
Likert Scales and Data Analyses
李克特式量表和数据分析
by I. Elaine Allen and Christopher A. Seaman
作者：I. Elaine Allen 和 Christopher A. Seaman
Surveys are consistently used to measure quality. For example, surveys might be used to gauge customer perception of product quality or quality performance in service delivery.
我们一贯采用调查的方法来衡量质量。比如说，顾客感知的产品质量或服务质量都可以用调查的方法来衡量。
Likert scales are a common ratings format for surveys. Respondents rank quality from high to low or best to worst using five or seven levels.
李克特式量表是一种常见的调查表评级格式，参与调查评级的受访者按照质量从高到低或者从好到坏分为五到七个档次。
Statisticians have generally grouped data collected from these surveys into a hierarchy of four levels of measurement:
统计人员一般将这些通过调查收集的数据按照4种测量方式进行分类。
Data analyses using nominal, interval and ratio data are generally straightforward and transparent. Analyses of ordinal data, particularly as it relates to Likert or other scales in surveys, are not. This is not a new issue. The adequacy of treating ordinal data as interval data continues to be controversial in survey analyses in a variety of applied fields.
使用名义数据，区间数据和比率数据进行数据分析时一般都直截了当，一目了然；而进行有序数据分析时，尤其数据跟李克特量表或者其他调查表相关时，就不那么一目了然了。这并不是一个新问题，在不同的应用领域，恰当地将有序数据作为区间数据进行数据处理一直保持着争议。
An underlying reason for analyzing ordinal data as interval data might be the contention that parametric statistical tests (based on the central limit theorem) are more powerful than nonparametric alternatives. Also, conclusions and interpretations of parametric tests might be considered easier to interpret and provide more information than nonparametric alternatives.
把序列数据作为区间数据来进行分析的根本原因可能是这样的论点，即（根据中心极限定理）参数的统计验证比非参数变量的验证更有说服力。同时，一般都会认为参数验证的结论和解释要比非参数变量更容易说明并提供更多的信息。。
However, treating ordinal data as interval (or even ratio) data without examining the values of the dataset and the objectives of the analysis can both mislead and misrepresent the findings of a survey. To examine the appropriate analyses of scalar data and when its preferable to treat ordinal data as interval data, we will concentrate on Likert scales.
然而，将序列数据作为区间数据甚至说比例数据来处理，而不对数据的值的设置和分析的目的进行检测，可能会误导或歪曲调查结果。检验数据分析方法是否适用，看看有序数据是何时更适合作为区间数据进行数据处理，我们集中看一下李克特量表。
Basics of Likert Scales
李克特量表基本信息
Likert scales were developed in 1932 as the familiar five-point bipolar response that most people are familiar with today.3 These scales range from a group of categories—least to most—asking people to indicate how much they agree or disagree, approve or disapprove, or believe to be true or false. There’s really no wrong way to build a Likert scale. The most important consideration is to include at least five response categories. Some examples of category groups appear in Table 1.
李克特量表形成于1932年，其形式为五点双极响应，这一形式在现今仍为人们所熟知。这一量表的范围从少到多进行分类，调查过程中让人们按照同意或不同意，赞成或不赞成的程度或者认为是对或者是错的程度来进行回答。建立李克特量表没有对错之分，最重要的就是要考虑好如何进行至少5种响应的分类，表1中给出一些例子。
The ends of the scale often are increased to create a seven-point scale by adding “very” to the respective top and bottom of the five-point scales. The seven-point scale has been shown to reach the upper limits of the scale’s reliability.4 As a general rule, Likert and others recommend that it is best to use as wide a scale as possible. You can always collapse the responses into condensed categories, if appropriate, for analysis.
最终的调查表一般会在我们所见的五点调查表中的最上端和最下端增加“非常非常”的极端表述而创立“七点”调查表。七点调查表已经用来显示调查范围上限所能达到的信赖性。作为一般规则，李克特和其他人建议最好使用尽可能宽范围的量表，这样你可以在适宜时将其压缩至更集中的分类来分析。
With that in mind, scales are sometimes truncated to an even number of categories (typically four) to eliminate the “neutral” option in a “forced choice” survey scale. Rensis Likert’s original paper clearly identifies there might be an underlying continuous variable whose value characterizes the respondents’ opinions or attitudes and this underlying variable is interval level, at best.
考虑到这一点，量表有时会变成偶数分类（一般是4项），消除“中立”选项制作成“强制选择”的调查表，Rensis Likert最初在论文中清晰地表明可能存在一个隐含的连续变量，该变量的值能够描述被调查者的观点或态度，这个变量充其量能达到区间数据的水平。
Analysis, Generalization To Continuous Indexes
分析，扩展到连续性数据
As a general rule, mean and standard deviation are invalid parameters for descriptive statistics whenever data are on ordinal scales, as are any parametric analyses based on the normal distribution. Nonparametric procedures—based on the rank, median or range—are appropriate for analyzing these data, as are distribution free methods such as tabulations, frequencies, contingency tables and chi-squared statistics.
作为一般的规则，描述性的统计数据使用序列量表的时候，平均值和标准差是不可用的，同样的基于正态分布的其他参数也是不可用的。基于序列，中位数或极差的非参数程序适于分析这一的数据，其他的不受分布形态影响的分析方法如表格，频率，应变表或者卡方统计也是可行的。
Kruskall-Wallis models can provide the same type of results as an analysis of variance, but based on the ranks and not the means of the responses. Given these scales are representative of an underlying continuous measure, one recommendation is to analyze them as interval data as a pilot prior to gathering the continuous measure.
Kruskall-Wallis模型能够能够提供与方差分析同类的结果，但仅仅是基于序列数据而不是响应的平均值。既然这一量表是代表一个潜在的连续量值，一个建议是分析这些区间数据，并作为收集连续数据之前的一个指引。
Table 2 includes an example of misleading conclusions, showing the results from the annual Alfred P. Sloan Foundation survey of the quality and extent of online learning in the United States. Respondents used a Likert scale to evaluate the quality of online learning compared to face-to-face learning.
表2中给出一个误导性结论的例子，表中给出的是艾尔弗雷德斯隆基金会一年一度关于美国在线学习质量和范围的调查结果，被调查者使用李克特量表来评估在线学习的质量并和面对面学习的质量进行对比。
表2 李克特量表样本
While 60%-plus of the respondents perceived online learning as equal to or better than face-to-face, there is a persistent minority that perceived online learning as at least somewhat inferior. If these data were analyzed using means, with a scale from 1 to 5 from inferior to superior, this separation would be lost, giving means of 2.7, 2.6 and 2.7 for these three years, respectively. This would indicate a slightly lower than average agreement rather than the actual distribution of the responses.
虽然60%以上的被调查者认为在线学习的效果等同于或者好于面对面学习的效果，但是一直有一部分人认为在线学习的效果要略逊一筹，如果这部分数据用平均值的方法进行分析，从优到劣共分为五个等级范围的话，这一部分的数据就会丢失，近三年来的平均值分别为2.7,2.6,2,7，这会表示一个略低于平均水平的认可，而不是实际的响应的分布。
A more extreme example would be to place all the respondents at the extremes of the scale, yielding a mean of “same” but a completely different interpretation from the actual responses.
一个更极端的例子是把所有的被调查者都放在极端的调查范围内，调查的平均值也会相同，但是调查的结果跟实际的响应就完全不同了。
Under what circumstances might Likert scales be used with interval procedures? Suppose the rank data included a survey of income measuring $0, $25,000, $50,000, $75,000 or $100,000 exactly, and these were measured as “low,” “medium” and “high.”
在什么情况下能够使用带有区间程序李克特量表呢？比如说对于收入的调查进行排序，分为$0, $25,000, $50,000, $75,000 or $100,000，这些数据可以描述为“低”“中”“高”。
The “intervalness” here is an attribute of the data, not of the labels. Also, the scale item should be at least five and preferably seven categories.
这里的“区间”是数据的属性，而不是数据的标签，同时，范围区分项至少要有五项，而更推荐的是七项。
Another example of analyzing Likert scales as interval values is when the sets of Likert items can be combined to form indexes. However, there is a strong caveat to this approach: Most researchers insist such combinations of scales pass the Cronbach’s alpha or the Kappa test of intercorrelation and validity.
将李克特量表作为区间值的另一个例子是李克特量表的项目能够集合成带有标志性的度量标准，当然这需要足够的说明才能实现：大多数调查者坚持这样的集合量表要通过Cronbach的α检验或Kappa检验来确认其交互性和有效性。
Also, the combination of scales to form an interval level index assumes this combination forms an underlying characteristic or variable.
此外，度量范围的有机结合形成具有标志性的度量标准假设这些组合形成了基本的变量或特征。
Alternative Continuous Measures for Scales
量表的可选择连续性测量方法
Alternatives to using a formal Likert scale can be the use of a continuous line or track bar. For pain measurement, a 100 mm line can be used on a paper survey to measure from worst ever to best ever, yielding a continuous interval measure.连续性线或轨迹栏来可作为李克特量表的替换方案。比如疼痛的检测，可以在纸上画100mm的线来记录最疼和最轻，从而来得到连续测量的区间间隔。
In the advent of many online surveys, this can be done with track bars similar to those illustrated in Figure 1. The respondents here can calibrate their responses to continuous intervals that can be captured by survey software as continuous values.
许多在线调查的结果就能用如图1所示的轨道栏来实现，这里被调查者能通过调查软件拖动鼠标来确认他们的响应值。
Conclusion
结论
Your initial analysis of Likert scalar data should not involve parametric statistics but should rely on the ordinal nature of the data. While Likert scale variables usually represent an underlying continuous measure, analysis of individual items should use parametric procedures only as a pilot analysis.
你最初的李克特量表数据分析可能并不包括参数统计量，但是却基于序列的自然数据，而李克特量表的变量通常表征为对一个持续的基本变量的测量，对于单独项目的分析，使用参数程序只能作为实验性分析。
Combining Likert scales into indexes adds values and variability to the data. If the assumptions of normality are met, analysis with parametric procedure can be followed. Finally, converting a five or seven category instrument to a continuous variable is possible with a calibrated line or track bar.
把李克特量表结合成指数，会增加数据的价值，也会增加变数。如果符合正态验证的假设，那么就是用参数程序，最终，通过轨道栏或标尺将5项或7项分类的量表转化成连续的变量是可能的。
REFERENCES
参考文献
BIBLIOGRAPHY
参考书目
I. ELAINE ALLEN is an associate professor of statistics and entrepreneurship at Babson College in Babson Park, MA. She has a doctorate in statistics from Cornell University in Ithaca, NY. Allen is a senior member of ASQ.
I. ELAINE ALLEN 是巴布森大学统计学和创业学的副教授，她获得了纽约伊萨卡康奈尔大学的统计学博士学位，是美国质量学会的资深会员。
CHRISTOPHER A. SEAMAN is a doctoral student in mathematics at the Graduate Center of City University of New York.
CHRISTOPHER A. SEAMAN是纽约城市大学研究中心数学统计专业的博士研究生。
对下面感兴趣的朋友，请留下你的预计完成时间和邮箱，并发送信息给小编D，谢谢~
本文翻译者：xy_persist 校稿者： liphking
Likert Scales and Data Analyses
李克特式量表和数据分析
by I. Elaine Allen and Christopher A. Seaman
作者：I. Elaine Allen 和 Christopher A. Seaman
Surveys are consistently used to measure quality. For example, surveys might be used to gauge customer perception of product quality or quality performance in service delivery.
我们一贯采用调查的方法来衡量质量。比如说，顾客感知的产品质量或服务质量都可以用调查的方法来衡量。
Likert scales are a common ratings format for surveys. Respondents rank quality from high to low or best to worst using five or seven levels.
李克特式量表是一种常见的调查表评级格式，参与调查评级的受访者按照质量从高到低或者从好到坏分为五到七个档次。
Statisticians have generally grouped data collected from these surveys into a hierarchy of four levels of measurement:
统计人员一般将这些通过调查收集的数据按照4种测量方式进行分类。
- Nominal data: The weakest level of measurement representing categories without numerical representation.
- 名义数据：最低水平的测量数据分类方法，没有数值表示。
- Ordinal data: Data in which an ordering or ranking of responses is possible but no measure of distance is possible.
- 有序数据：数据可能按照某一顺序或者排序进行反馈，但是可能没有测量数据间隔。
- Interval data: Generally integer data in which ordering and distance measurement are possible.
- 区间数据：数据一般为整数，数据的排序或间隔都进行测量。
- Ratio data: Data in which meaningful ordering, distance, decimals and fractions between variables are possible.
- 比率数据： 数据中可以包含变量的有意义的排顺、间隔、小数和分数
Data analyses using nominal, interval and ratio data are generally straightforward and transparent. Analyses of ordinal data, particularly as it relates to Likert or other scales in surveys, are not. This is not a new issue. The adequacy of treating ordinal data as interval data continues to be controversial in survey analyses in a variety of applied fields.
使用名义数据，区间数据和比率数据进行数据分析时一般都直截了当，一目了然；而进行有序数据分析时，尤其数据跟李克特量表或者其他调查表相关时，就不那么一目了然了。这并不是一个新问题，在不同的应用领域，恰当地将有序数据作为区间数据进行数据处理一直保持着争议。
An underlying reason for analyzing ordinal data as interval data might be the contention that parametric statistical tests (based on the central limit theorem) are more powerful than nonparametric alternatives. Also, conclusions and interpretations of parametric tests might be considered easier to interpret and provide more information than nonparametric alternatives.
把序列数据作为区间数据来进行分析的根本原因可能是这样的论点，即（根据中心极限定理）参数的统计验证比非参数变量的验证更有说服力。同时，一般都会认为参数验证的结论和解释要比非参数变量更容易说明并提供更多的信息。。
However, treating ordinal data as interval (or even ratio) data without examining the values of the dataset and the objectives of the analysis can both mislead and misrepresent the findings of a survey. To examine the appropriate analyses of scalar data and when its preferable to treat ordinal data as interval data, we will concentrate on Likert scales.
然而，将序列数据作为区间数据甚至说比例数据来处理，而不对数据的值的设置和分析的目的进行检测，可能会误导或歪曲调查结果。检验数据分析方法是否适用，看看有序数据是何时更适合作为区间数据进行数据处理，我们集中看一下李克特量表。
Basics of Likert Scales
李克特量表基本信息
Likert scales were developed in 1932 as the familiar five-point bipolar response that most people are familiar with today.3 These scales range from a group of categories—least to most—asking people to indicate how much they agree or disagree, approve or disapprove, or believe to be true or false. There’s really no wrong way to build a Likert scale. The most important consideration is to include at least five response categories. Some examples of category groups appear in Table 1.
李克特量表形成于1932年，其形式为五点双极响应，这一形式在现今仍为人们所熟知。这一量表的范围从少到多进行分类，调查过程中让人们按照同意或不同意，赞成或不赞成的程度或者认为是对或者是错的程度来进行回答。建立李克特量表没有对错之分，最重要的就是要考虑好如何进行至少5种响应的分类，表1中给出一些例子。
The ends of the scale often are increased to create a seven-point scale by adding “very” to the respective top and bottom of the five-point scales. The seven-point scale has been shown to reach the upper limits of the scale’s reliability.4 As a general rule, Likert and others recommend that it is best to use as wide a scale as possible. You can always collapse the responses into condensed categories, if appropriate, for analysis.
最终的调查表一般会在我们所见的五点调查表中的最上端和最下端增加“非常非常”的极端表述而创立“七点”调查表。七点调查表已经用来显示调查范围上限所能达到的信赖性。作为一般规则，李克特和其他人建议最好使用尽可能宽范围的量表，这样你可以在适宜时将其压缩至更集中的分类来分析。
With that in mind, scales are sometimes truncated to an even number of categories (typically four) to eliminate the “neutral” option in a “forced choice” survey scale. Rensis Likert’s original paper clearly identifies there might be an underlying continuous variable whose value characterizes the respondents’ opinions or attitudes and this underlying variable is interval level, at best.
考虑到这一点，量表有时会变成偶数分类（一般是4项），消除“中立”选项制作成“强制选择”的调查表，Rensis Likert最初在论文中清晰地表明可能存在一个隐含的连续变量，该变量的值能够描述被调查者的观点或态度，这个变量充其量能达到区间数据的水平。
Analysis, Generalization To Continuous Indexes
分析，扩展到连续性数据
As a general rule, mean and standard deviation are invalid parameters for descriptive statistics whenever data are on ordinal scales, as are any parametric analyses based on the normal distribution. Nonparametric procedures—based on the rank, median or range—are appropriate for analyzing these data, as are distribution free methods such as tabulations, frequencies, contingency tables and chi-squared statistics.
作为一般的规则，描述性的统计数据使用序列量表的时候，平均值和标准差是不可用的，同样的基于正态分布的其他参数也是不可用的。基于序列，中位数或极差的非参数程序适于分析这一的数据，其他的不受分布形态影响的分析方法如表格，频率，应变表或者卡方统计也是可行的。
Kruskall-Wallis models can provide the same type of results as an analysis of variance, but based on the ranks and not the means of the responses. Given these scales are representative of an underlying continuous measure, one recommendation is to analyze them as interval data as a pilot prior to gathering the continuous measure.
Kruskall-Wallis模型能够能够提供与方差分析同类的结果，但仅仅是基于序列数据而不是响应的平均值。既然这一量表是代表一个潜在的连续量值，一个建议是分析这些区间数据，并作为收集连续数据之前的一个指引。
Table 2 includes an example of misleading conclusions, showing the results from the annual Alfred P. Sloan Foundation survey of the quality and extent of online learning in the United States. Respondents used a Likert scale to evaluate the quality of online learning compared to face-to-face learning.
表2中给出一个误导性结论的例子，表中给出的是艾尔弗雷德斯隆基金会一年一度关于美国在线学习质量和范围的调查结果，被调查者使用李克特量表来评估在线学习的质量并和面对面学习的质量进行对比。
表2 李克特量表样本
While 60%-plus of the respondents perceived online learning as equal to or better than face-to-face, there is a persistent minority that perceived online learning as at least somewhat inferior. If these data were analyzed using means, with a scale from 1 to 5 from inferior to superior, this separation would be lost, giving means of 2.7, 2.6 and 2.7 for these three years, respectively. This would indicate a slightly lower than average agreement rather than the actual distribution of the responses.
虽然60%以上的被调查者认为在线学习的效果等同于或者好于面对面学习的效果，但是一直有一部分人认为在线学习的效果要略逊一筹，如果这部分数据用平均值的方法进行分析，从优到劣共分为五个等级范围的话，这一部分的数据就会丢失，近三年来的平均值分别为2.7,2.6,2,7，这会表示一个略低于平均水平的认可，而不是实际的响应的分布。
A more extreme example would be to place all the respondents at the extremes of the scale, yielding a mean of “same” but a completely different interpretation from the actual responses.
一个更极端的例子是把所有的被调查者都放在极端的调查范围内，调查的平均值也会相同，但是调查的结果跟实际的响应就完全不同了。
Under what circumstances might Likert scales be used with interval procedures? Suppose the rank data included a survey of income measuring $0, $25,000, $50,000, $75,000 or $100,000 exactly, and these were measured as “low,” “medium” and “high.”
在什么情况下能够使用带有区间程序李克特量表呢？比如说对于收入的调查进行排序，分为$0, $25,000, $50,000, $75,000 or $100,000，这些数据可以描述为“低”“中”“高”。
The “intervalness” here is an attribute of the data, not of the labels. Also, the scale item should be at least five and preferably seven categories.
这里的“区间”是数据的属性，而不是数据的标签，同时，范围区分项至少要有五项，而更推荐的是七项。
Another example of analyzing Likert scales as interval values is when the sets of Likert items can be combined to form indexes. However, there is a strong caveat to this approach: Most researchers insist such combinations of scales pass the Cronbach’s alpha or the Kappa test of intercorrelation and validity.
将李克特量表作为区间值的另一个例子是李克特量表的项目能够集合成带有标志性的度量标准，当然这需要足够的说明才能实现：大多数调查者坚持这样的集合量表要通过Cronbach的α检验或Kappa检验来确认其交互性和有效性。
Also, the combination of scales to form an interval level index assumes this combination forms an underlying characteristic or variable.
此外，度量范围的有机结合形成具有标志性的度量标准假设这些组合形成了基本的变量或特征。
Alternative Continuous Measures for Scales
量表的可选择连续性测量方法
Alternatives to using a formal Likert scale can be the use of a continuous line or track bar. For pain measurement, a 100 mm line can be used on a paper survey to measure from worst ever to best ever, yielding a continuous interval measure.连续性线或轨迹栏来可作为李克特量表的替换方案。比如疼痛的检测，可以在纸上画100mm的线来记录最疼和最轻，从而来得到连续测量的区间间隔。
In the advent of many online surveys, this can be done with track bars similar to those illustrated in Figure 1. The respondents here can calibrate their responses to continuous intervals that can be captured by survey software as continuous values.
许多在线调查的结果就能用如图1所示的轨道栏来实现，这里被调查者能通过调查软件拖动鼠标来确认他们的响应值。
Conclusion
结论
Your initial analysis of Likert scalar data should not involve parametric statistics but should rely on the ordinal nature of the data. While Likert scale variables usually represent an underlying continuous measure, analysis of individual items should use parametric procedures only as a pilot analysis.
你最初的李克特量表数据分析可能并不包括参数统计量，但是却基于序列的自然数据，而李克特量表的变量通常表征为对一个持续的基本变量的测量，对于单独项目的分析，使用参数程序只能作为实验性分析。
Combining Likert scales into indexes adds values and variability to the data. If the assumptions of normality are met, analysis with parametric procedure can be followed. Finally, converting a five or seven category instrument to a continuous variable is possible with a calibrated line or track bar.
把李克特量表结合成指数，会增加数据的价值，也会增加变数。如果符合正态验证的假设，那么就是用参数程序，最终，通过轨道栏或标尺将5项或7项分类的量表转化成连续的变量是可能的。
REFERENCES
参考文献
- Gideon Vigderhous, “The Level of Measurement and ‘Permissible’ Statistical Analysis in Social Research,” Pacific Sociological Review, Vol. 20, No. 1, 1977, pp. 61-72.
- Gideon Vigderhous, 社会研究领域“允许”统计分析的测量水准，太平洋社会观察，Vol. 20, No. 1, 1977, pp. 61-72.
- Ulf Jakobsson, “Statistical Presentation and Analysis of Ordinal Data in Nursing Research,” Scandinavian Journal of Caring Sciences, Vol. 18, 2004, pp. 437-440.
- Ulf Jakobsson, 护理研究中有序数据的统计描述和分析，斯堪的纳维亚关怀科学杂质，Vol. 18, 2004, pp. 437-440.
- Rensis Likert, “A Technique for the Measurement of Attitudes,” Archives of Psychology, 1932, Vol. 140, No. 55.
- Rensis Likert, 一项态度测量的技术，心理学，1932, Vol. 140, No. 55.
- Jum C. Nunnally, Psychometric Theory, McGraw Hill, 1978.
- Jum C. Nunnally, 心理测量原理，McGraw Hill, 1978.
- Dennis L. Clasen and Thomas J. Dormody, “Analyzing Data Measured by Individual Likert-Type Items,” Journal of Agricultural Education, Vol. 35, No. 4, 1994.
- Dennis L. Clasen and Thomas J. Dormody, 特定的李克特量表项目进行分析数据测量，农业教育，Vol. 35, No. 4, 1994.
BIBLIOGRAPHY
参考书目
- Jacoby, Jacob, and Michael S. Matell, “Three-Point Likert Scales Are Good Enough,” Journal of Marketing Research, Vol. 8, No. 4, 1971, pp. 495-500.
- Jacoby, Jacob, and Michael S. Matell, 三项李克特量表就足够了，市场研究，Vol. 8, No. 4, 1971, pp. 495-500.
- Jamieson, Susan, “Likert Scales: How to (Ab)use Them,” Medical Education, Vol. 38, No. 12), 2004, pp. 1,217-1,218.
- Jamieson, Susan, 李克特量表：怎么使用/滥用该量表，医学教育，Vol. 38, No. 12), 2004, pp. 1,217-1,218.
I. ELAINE ALLEN is an associate professor of statistics and entrepreneurship at Babson College in Babson Park, MA. She has a doctorate in statistics from Cornell University in Ithaca, NY. Allen is a senior member of ASQ.
I. ELAINE ALLEN 是巴布森大学统计学和创业学的副教授，她获得了纽约伊萨卡康奈尔大学的统计学博士学位，是美国质量学会的资深会员。
CHRISTOPHER A. SEAMAN is a doctoral student in mathematics at the Graduate Center of City University of New York.
CHRISTOPHER A. SEAMAN是纽约城市大学研究中心数学统计专业的博士研究生。
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