Climate models are powerful tools used for investigating how the climate system works, for making scenarios of the future climate and for assessing potential impacts of climatic changes . Using a numerical representation of the real climate system, climate models are designed as systems of complex differential equations based on physical, biological, and chemical principles. In the virtual world of climate models, scientists can perform experiments that are not feasible in the real world climate system. For example, one can neglect or simplify all but one process, in order to identify the role of this particular process clearly, for example, the influence of changes in solar irradiance on the radiative properties of the atmosphere, or to test hypotheses related to this process. In an analogous fashion, the overall effect of several processes, acting jointly, can be investigated.

In order to assess the magnitude of the effects of the processes in question on the climate, it is often convenient to analyse their impact on the radiative balance of the Earth . The net change in the Earth's radiative balance at the tropopause (incoming energy flux minus outgoing energy flux expressed in watts per square metre (W m-2)) caused by a change in a climate driver is called a radiative forcing (see, for example, ). Sometimes scientists use the term climate forcing instead of radiative forcing . We will generally do so in our discussion, or we simply write just forcing.

Examples of external natural drivers of climate change, which are capable of inducing climate forcing, are changes in solar radiation, changes in the orbital parameters of the Earth, and volcanic eruptions. There are also external drivers of climate change of anthropogenic origin, for example, the ongoing release of carbon dioxide to the atmosphere, primarily by burning fossil fuels, the emissions of aerosols through various industrial and burning processes, and changes in land use . Climate variability can also be caused by various processes internal to the climate system itself . Ocean and atmosphere circulation and their variations and mutual interactions are examples of processes that are clearly internal to the climate system.

The range of types of climate models is very wide. Here, our focus is on the most sophisticated state-of-the art climate models referred to as global climate models (GCMs) or Earth system models (ESMs). As computing capabilities have evolved during the past decades, the complexity of GCMs and ESMs has substantially increased: for instance, the number of components of the Earth system that can be included and coupled in GCMs and ESMs has increased, or the previously often performed equilibrium simulations can now be replaced by transient simulations, driven, for example, by temporal changes in the atmospheric greenhouse gases (GHGs) and aerosol loading (see, for example, ).

However, despite great advances achieved during the past decades, some simplifications are unavoidable, for example, due to the range of temporal and spatial scales involved and/or incomplete knowledge about some processes. As a consequence, the complexity of the most sophisticated climate models is still far from the complexity of the real climate system. Even a careful design cannot guarantee that each component of climate modelling, for example, parameterisation of subgrid-scale processes, has been employed in its optimal form. Also, our knowledge about various feedback processes, for example, cloud feedback , that may either amplify or damp the direct effect of a given forcing is not complete.

Another type of complication is that forcing reconstructions too may be uncertain. As examples, uncertainties can be large for such anthropogenic forcings as aerosol forcing and land use forcing , and the amplitude of natural solar irradiance changes in the last millennium has been disputed .

All the above-mentioned issues together point naturally to the importance of carefully undertaking evaluation of climate model simulations by comparison against the observed climate state and variability. An important role in this context has been played by so-called detection and attribution (D&A) studies (e.g., ). Within these studies, the question of attribution of observed climate change to real-world forcings is addressed simultaneously with the question of consistency between simulated and observed climate change. That is, one of the goals of D&A studies is to evaluate the ability of forced climate models to simulate observed climate change correctly.

Based on the ideas of , statistical methods used in D&A studies performed to date have been given different representations and typically are referred to as “optimal fingerprinting” techniques. In the present work, the focus is on the representation associated with linear regression models, described by .

In terms of the near-surface temperature, which has been a climatic variable of interest in many D&A studies, an advantageous assumption made in is that neither the real temperature responses to particular forcings, nor the simulated responses to imposed forcings obtained in experiments with complex GCMs or ESMs are directly observable; that is, they are latent. This has motivated the use of regression models with latent variables where both explanatory and response variables are contaminated with noise. In the statistical literature, these regression models are known as measurement error (ME) models (sometimes also-called errors-in-variables models).

Being the simplest model among statistical models with latent variables, the ME model specification has proved to be a useful tool within many D&A studies that greatly contributed to the understanding of the causes of climate variability. However, as recognised by several researchers, this statistical model is associated with certain limitations, for example, the inability to take into account the effects of possible interactions between forcings (see, for example, ) or the inability to account for non-climatic noise in the observational data.

Within the present work, observational data are defined as data consisting of instrumental temperature measurements, when they are available, and temperature reconstructions derived from climate proxy data, i.e. from indirect climate information from various natural archives such as tree rings, lake sediments, and cave speleothems.

A general statistical model, used for assessing the individual contribution of m forcings in many D&A studies made after 2003, is given by the following ME model with a vector explanatory variable (see, for example, ): 1yg=∑i=1mβi(xig-νig)+ν0g, where yg is the mean-centred observed/reconstructed temperature, where the index g reflects the fact that the {yg} sequence is allowed to be a temperature field, arrayed in space and in time; xig is the mean-centred simulated temperature, generated by the climate model forced only by a reconstruction of forcing i; νig is the simulated internal temperature variability associated with the climate model driven by forcing i; ν0g is the real internal temperature variability, i.e. the real latent unforced component embedded in yg; xig-νig is the simulated latent temperature response to forcing i, embedded in xig; and βi is a scaling factor associated with forcing i.

For the purpose of our analysis let us rewrite model () with respect to the specified forcings and partly by using our notations. Moreover, realising that the ME model in the form presented does not take into account the initial space-time dimensionality of data but simply presupposes that each model variable is associated with a single vector of values of the same length across all variables, we replace the subindex g by the subindex t, representing the time point t, t=1,2,…,n. So for the time point t and any region, we get the following model: yt=βSolξSol tS+βOrbξOrb tS+βVolcξVolc tS+βLandξLand tS+βGHGξGHG tS+ν0t,2xft=ξftS+νft, where f ranges over {Sol,Orb,Volc,Land,GHG}. In statistical literature, this kind of a ME model is known as a ME model with no error in the equation see, for example,. To enable the inferences, one typically assumes that the error vectors  (ν0t,νft)′ are normally and independently distributed under a wide range of distributional assumptions for ξftS. Also, one assumes that the errors are uncorrelated with ξftS, associated in their turn with a non-singular variance–covariance matrix.

In the case of Eq. (), the term “no error in the equation” refers to the fact that the overall true temperature response to all forcings, acting in the real-world climate system and embedded in yt, is modelled as an error-free linear function of the five simulated temperature responses of interest, that is 3ξALLtT=∑fβf⋅ξftS, where f ranges over {Sol,Orb,Volc,Land,GHG}, and ξALLtT is our notation for the overall true temperature response to all forcings acting in the real-world climate system. From Eq. (), it also follows that for each individual forcing f, 4ξftT=βf⋅ξftS, where ξftT is a real-world counterpart to ξftS.

Equations () and () are justified by the assumption, made in D&A studies, that the (large-scale) shape of the true temperature response is correctly simulated by the climate model under consideration (see, for example, ). Hence, we may say that model () assumes that the simulated temperature response ξftS can differ from the corresponding ξftT only in its magnitude, represented by the parameter βf.

The introduction of an error into Eq. () would entail another structure of the variable representing the random variability in yt. In other words, ν0t would only be a part of the random component of yt. The structure of the random component may be even more complex if one wishes to take into account a non-climatic noise, which can constitute quite a large part of temperatures reconstructed from proxy data and which can also exist in varying amounts in instrumental temperature observations . According to , the main reason for excluding the corresponding error term, known as observation error, is that its autocorrelation structure is assumed to differ from that associated with the internal temperature variability.

From the estimation point of view, a consequence of adding various error terms to ν0 is that one may need to use another estimator of βfs instead of that used in D&A studies. The estimator used is known as the total least squares (TLS) estimator, and it requires the knowledge of the ratios of the variances of νf and ν0, which can be quite challenging to derive if ν0 is replaced by a multi-component variable.

Importantly, the TLS estimator remains unchanged if the whole error variance–covariance matrix is known a priori. In practice, this knowledge permits us to check for model validity, provided one derives this a priori knowledge from a source independent of the sample variance–covariance matrix of the observed variables. In D&A studies, such a source is unforced (control) climate model simulations.

We would also like to emphasise the fact that the TLS estimator is obtained under the condition that all latent variables are correlated. This entails that if some ξfSs are highly mutually correlated or at least one of ξfSs does not vary much, unreliable estimates are expected. Thus, it is important to check whether the variance–covariance matrix of the latent variables is singular or not.

The main parameters of interest, estimated within D&A studies, are the coefficients βf. They are involved in two important hypotheses. The first one concerns the detection of the corresponding simulated temperature response in the observed climate record yt. The rejection of H0: βf=0 indicates that ξfS is detected in yt. The rejection also indicates the so-called “strong-signal regime” , associated with high reliability of all the parameter estimates and their associated confidence intervals (provided, of course, the variance–covariance matrix of latent variables is still non-singular).

The detection of a simulated temperature response, however, in the observed climate record is not sufficient for attributing this detected simulated temperature signal to the corresponding real-world forcing f. For this, one needs to show that ξftS is consistent with ξftT, i.e., that Eq. () holds with βf=1. Thus, the second hypothesis of interest in D&A studies is H0: βf=1, known as the hypothesis of consistency. However, as emphasised by , the test for consistency, referred to by the authors as “attribution consistency test”, does not constitute a complete attribution assessment, though it contributes important evidence to such an assessment.

Another important feature of model () is that it does not take into account the effect of possible interaction(s) between forcings. Instead, the model assumes the additivity of forcing effects. The issue of additivity has been recognised and discussed by many researchers within the D&A field. Several analyses have been performed to investigate the significance of interactions in simulated climate systems (see, for example, ). The results seem to support the assumption of additivity when it concerns temperature. Nevertheless, it would be beneficial to develop a statistical model permitting a simultaneous assessment of the magnitude of both individual and interaction effects of forcings on temperature.

Summarising the overview above, we may say that it is highly motivating to investigate possibilities of extending the ME model representation to more complex statistical models in order to overcome the limitations of the ME model.

To achieve this aim, we suggest using the close link between ME, CFA, and SEM models. To see that the ME model in Eq. () can be viewed as a CFA model, let us first rewrite this model in the matrix form as shown in Eq. (). 5

In the matrix form given in Eq. (), the ME model is represented as an unstandardised CFA model, that is, a CFA model with unstandardised latent factors. The pattern of the model coefficients, also-called factor loadings in CFA literature, reflects our conviction that observations on xf were generated by the climate model driven only by forcing f. In other words, model () tells us that each xft only has one common latent factor with yt, that is ξftS.

Note also that an unstandardised factor model is associated with an unstandardised solution. However, a standardised solution is preferred because its model coefficients (hereafter called standardised coefficients) make it possible to judge the relative importance of latent factors. The standardisation of the latent factors in model () is accomplished by standardising their variances to 1. The resulting CFA model is given in Eq. (). Since the model has been derived from a ME model, we call model () a six-indicator and five-factor ME-CFA model, abbr. ME-CFA(6, 5) model.

In model (), the ratio κf/αf gives us back βf, associated with the ME model in Eq. (), for each forcing f. Thus, the hypotheses concerning βf are applicable to the parameters κf and αf. In particular, the hypothesis H0: βf=0 corresponds to H0: κf=0, while the hypothesis of consistency is equivalent to H0: κf/αf=1 or equivalently H0: κf=αf.

In practice, it is not difficult to test various equality constraints within a CFA model – one simply fits the associated factor model under the constraints of interest. That is, one fits and tests simultaneously (for more theoretical details about the estimation of CFA models, see the Appendix). In the same manner, one may introduce restrictions on the correlations among the latent factors. 6 where ξ̃ftS=ξftS/σf2, αf=σf2, and , κf=βf⋅σf2 for each forcing f.

Another advantage of thinking in the spirit of CFA is that the factor model specification makes it possible to take into account the lack of additivity, which may arise due to possible interactions between forcing. This can be accomplished by adding observable variables associated with various multi-forcing climate model simulations (provided such simulations are available). Finally, this model specification seems to permit a complete attribution assessment, provided one uses ξfTs as common factors instead of ξfSs.

In this section, our aim is to formulate a basic CFA model with respect to the five specified forcings. The model is called basic because it is supposed to be modified depending on the climate-relevant characteristics of the specified forcings for the region and period of interest. Note also that we focus on a CFA model with standardised latent factors in order to enable meaningful comparisons of estimated effects of latent factors. For a brief account of general CFA models and the associated definitions, used in the following sections, see the Appendix.

As mentioned in the Introduction, one of the starting points for our framework is the SUN12 framework. Within the confines of the present work, we combine some of the definitions in SUN12 that are relevant for our work with our definitions.

Like D&A studies, decomposing the climate variability within the ME model into forced and unforced components, SUN12 also implements the same decomposition within its statistical model. However, unlike D&A studies, SUN12 allows more complex structures of random components both in yt and in xf. Using this feature of SUN12, our initial model for the mean-centred yt for a given region and time t, t=1,…,n, is as follows: 7yt=ξALLtT+ν̃t, where the forced component ξALLtT was defined in Eq. (), and the random component ν̃t, assumed to be independent of ξALLtT, contains, for each t, (1) the internal random variability of the real-world climate system, including any random variability due to the presence of the forcings, and (2) the non-climatic noise. Although SUN12 allows the non-climatic variability to vary depending on the time period, in this work, for simplicity, we will assume that the variance of ν̃t is constant for each time point.

Since there are five forcings under investigation, the next step is to rewrite yt in Eq. () under the assumption that only these five true forcings may systematically contribute to the variability in yt. In doing so, we also want to allow a term for the lack of additivity, due to possible interactions between the forcings. The resulting equation is as follows: 8yt=Strue⋅ξSoltT+Otrue⋅ξOrbtT+Vtrue⋅ξVolctT+Ltrue⋅ξLandtT+Gtrue⋅ξGHGtT+Itrue⋅ξinteracttT+ζALL tT+ν̃t︸=νt, where (i) ξinteractT denotes the overall temperature response to all possible interactions between the forcings under consideration; (ii) the parameters Strue, Otrue, and so on are standardised coefficients, denoting the magnitude of the individual contribution from the associated real-world forced processes to the variability in yt; and (iii) ζALLtT, assumed to be independent of all ξtTs in the equation, represents residual variability arising after extracting the six specified true temperature responses from ξALL tT. As one can see, the random component νt in Eq. () has a more complex structure than the corresponding random component ν0t in model Eq. ().

In a similar way, our initial model for xft is given by 9xft=ξftS+δ̃ft, where ξftS was defined in Sect. 2, and δ̃ft, assumed to be independent of ξftS, represents the simulated internal random temperature variability, including any random variability due to the presence of the forcing f. Hence, Eq. () does not require the same random variability for forced climate model simulations generated by the climate model under consideration when it is driven by different forcings.

The next step is to rewrite each xft as a function of the corresponding ξftT. The idea of modelling true temperature responses as common factors for xft and yt was suggested in SUN12 but without exploring its consequences. Here, it should be realised that the replacement of ξfSs by corresponding ξfTs is not only a question of different notation. This replacement entails different structures and interpretations of error terms, called specific factors in CFA literature. As a result, different factor models arise.

Let us describe this process by example of xSolt. The extraction of ξSol tT from ξSoltS leads to the following equation: 10xSolt=Ssim⋅ξSol tT+ζSol tS+δ̃Solt︸=δSolt, where the residual term ζSol tS, assumed to be independent of ξSol tT, arises as a result of extracting ξSoltT from ξSoltS and is supposed to represent the large-scale shape of ξSoltS. Further, the coefficient Ssim reflects the idea that the magnitude of the common factor ξSoltT, extracted from ξSoltS, is not necessarily the same as its true magnitude, represented by the coefficient Strue in Eq. ().

Assuming that both magnitude and the large-scale shape of the true temperature response are correctly simulated, Eq. () transforms to 11xSolt=Strue⋅ξSol tT+δ̃Solt. Thus, the meaning of the consistency within our framework is that a simulated temperature response ξftS can be represented as an error-free function of the corresponding real-world temperature response ξftT with the same magnitude as that observed in yt.

An important feature of our framework is that it involves xcomb as an additional observable variable, assumed to contain the simulated counterparts to each ξfT and to ξinteractT. Our initial model for xcombt is 12xcombt=ξcombtS+δ̃combt, which after extracting the six true temperature responses from ξcombtS transforms to 13xcombt=Ssim⋅ξSoltT+Osim⋅ξOrbtT+Vsim⋅ξVolctT+Lsim⋅ξLandtT+Gsim⋅ξGHGtT+Isim⋅ξinteracttT+ζcombtS+δ̃combt︸=δcombt. Notice that each ξftT in Eq. () is associated with the same coefficient as that representing the magnitude of ξftT within xft (see, for example, Eq. ). These equalities are justified by the fact that the underlying ξfS, embedded in xf and in xcomb, has been generated by the same implementation of forcing f and that the same climate model is used in both cases. Statistically, this repeatedness of each ξfS allows us to treat them (and ξfTs as well) as a repeatable outcome of random variables, each of which is assumed to be normally and independently distributed with zero mean and its own variance.

It is also important to highlight that under the CFA (and ME) model specification, all common latent factors can be related to each other only through correlations. Climatologically, this corresponds to viewing all underlying forcings as physically independent processes not capable of causing changes in each other but giving rise to temperature responses that can be either mutually correlated or not.

Given the preliminaries above, we can finally formulate our basic CFA model that hypothesises the consistency between the simulated and true temperature responses. The parameters of the resulting CFA model with seven indicators and six common factors are given in Table . Importantly, the CFA model presented hypothesises not only the consistency, but also zero correlation between ξSolT, ξOrbT, and ξVolcT. Our motive for these zero correlations is that the forcings that generate these temperature responses are acting on different timescales and with different character of their temporal evolutions. It is thus reasonable to expect that their temperature responses will not demonstrate a similar temporal pattern. On the other hand, we find it difficult to hypothesise zero correlations between ξ interactT and ξSolT,  ξOrbT, and ξVolcT. Thus each of the last three responses is allowed to be correlated with ξ interactT.

Another feature of the model is that the variances of the specific factors δ̃fs, where f represents either one of the individual forcings or their combination, are treated as known a priori, i.e. as fixed parameters. To obtain an a priori estimate of each σδ̃f2, we suggest using the following estimator if the xf ensemble of interest contains at least two members or replicates: 14σδ̃f2=∑t=1n∑i=1kf(xfrepl.it-x¯f.t)2n(kf-1),kf≥2, where x¯f.t is the average of the observations at time t. In case it is climatologically justified to assume that the forcing of interest has a negligible influence on the temperature variability, one can use instead the sample variance of all nkf observations, i.e. 15σδ̃f2=∑t=1n∑i=1kf(xfrepl.it-x¯f..)2nkf-1, where x¯f.. is the average of all observations. In Sect. 6, we discuss the assumptions associated with the estimators above and demonstrate in practice how these assumptions may be checked using the ideas of CFA.

The advantage of treating σδ̃f2s as fixed parameters is that it enables the estimation, i.e. identifiability, of a CFA model without hypothesising the consistency for each ξfS and the uncorrelatedness between ξSolT, ξOrbT, and ξVolcT (in case the latter is climatologically motivated for the region and period of study). The resulting CFA model is given in Table . This CFA model is just-identified with 0 degrees of freedom. As a consequence, it is not possible (and unmotivated) to test for lack of fit to the data.

In contrast, the CFA model in Table  is over-identified with 9 degrees of freedom, which makes it meaningful to test whether the empirical data conform to the hypothesised latent structure. If all the estimates of this CFA model are admissible and interpretable, and if the model fits the data adequately both statistically or heuristically (see Eqs. – in the Appendix), then one may say that there is no reason to reject the hypotheses that both magnitude and shape of the simulated temperature response are correctly simulated by the climate model under consideration.

In addition, the estimates of the factor loadings can be used for assessing the contribution of the real-world forcings to the variability in yt. For example, provided that the parameter estimates of the CFA model in Table  are admissible and climatologically interpretable, the rejection of H0: Strue=0 indicates that the true temperature response to the real-world solar forcing is detected in observational data. That is, the attribution to the real-world forcings is possible, even if the hypothesis of consistency is relaxed.

Both above-presented CFA(7, 6) models can be modified by setting desirable and climatologically justified constraints on the parameters. For example, for testing that the effect of interactions is negligible, one needs to set Itrue (and Isim, depending on which CFA model is considered) to zero. Importantly, one also needs to set each of the five associated correlations to zero in order to avoid under-identifiability. As a result, one gains 6 additional degrees of freedom if the over-identified CFA model from Table  is analysed. In the case of the CFA model from Table , the zero constraints imposed makes the model over-identified with 7 degrees of freedom.

Similar constraints can be placed on the parameters associated with other common factors, if one expects negligible forcing effects for the region of interest. Otherwise, the estimation procedure may become unstable, which may lead to an inadmissible solution or even the failure to converge to a solution.

Another reason to modify the CFA models presented arises when instead of xLand climate model simulations only xLand (anthr) or xLand (natural) climate model simulations are available. When replacing the indicator xLand by xLand (anthr) or xLand (natural), one needs to replace the common factor ξLandT as well with ξLand (anthr)T and ξLand (natural)T, respectively.

Such replacements may change the latent structure of the model. In case only xLand (anthr) is available, it seems reasonable to let ξLand (anthr)T be uncorrelated with ξSolT, ξOrbT, ξVolcT, and ξinteractT. because anthropogenic changes in land use can be viewed as processes independent of the natural forcings.

In case ξLandT is replaced by ξLand (natur)T, it seems climatologically justified to let ξLand (natur)T be correlated with other common factors, provided one does not expect a negligible effect of natural changes in the Land forcing within the region and period of interest. In this case, to avoid under-identifiability, one may set all the parameters, associated with ξLand (natural)T, to zero. This would correspond to viewing natural changes in Land cover as an internal climate process, contributing to the temperature variability randomly.

Regardless of the latent structure hypothesised, it is important to emphasise that xf in all possible CFA models can represent the mean sequence as well. As known (e.g. ), averaging over replicates of the same type of forced model leads to a time series with an enhanced forced climate signal and a reduced effect of the internal temperature variability of the corresponding forced climate model. This may considerably contribute to the stability of the estimation procedure, especially when the forcing of interest is weak rather than strong. Given kf replicates of xf, the replacement of xft by x¯ft implies that σδ̃f2 is replaced by σδ̃f2/kf.

If the solution obtained is admissible and climatologically defensible, the overall model fit to the data can be assessed both statistically and heuristically. In case of rejecting the hypothesised model, it is important to realise that the rejection does not unambiguously point to any particular constraint as at fault.

We also present the CFA model from Table  graphically by means of a path diagram. This would simplify a movement from the CFA model specification to the SEM model specification and makes it easier to overview (complex) causal relationships within SEM models. To understand a path diagram, let us explain its symbols:

A one-headed arrow represents a causal relationship between two variables, meaning that a change in the variable at the tail of the arrow will result in a change in the variable at the head of the arrow (with all other variables in the diagram held constant). The former type of variable is referred to as exogenous (Greek: “of external origin”) or independent variables because their causes lie outside the path diagram. Variables that receive causal inputs in the diagram are referred to as endogenous (“of internal origin”) or dependent variables because their values are influenced by variables that lie within the path diagram.

The path diagram for the CFA model in Table  is depicted in Fig. .

The CFA model specification can also be used for assessing the overall forcing effect. For this purpose, we formulated a two-indicator one-factor CFA model, which we present in the Supplement, together with the corresponding ME model used in D&A studies.

In CFA, latent variables can be related to each other exclusively in terms of correlations, which says nothing about the underlying reasons for the correlation (association). Indeed, an association between two variables, say A and B, may arise because of the following:

A influences (or causes) B.

An example of physically complicated climatological relationships is climate–vegetation interactions. To reflect (to some extent) this climatological mechanism statistically, we recall first that ξLandtT is a two-component temperature response, containing ξLand (natur)tT and ξLand (anthr)tT. Further, we note that natural changes in the Land cover and vegetation are processes that physically depend on the solar, orbital, and volcanic forcings and their interactions. From the perspective of structural equation modelling this means that ξLand (natur)tT can be described as causally dependent on ξSoltT, ξOrbtT, ξVolctT, and ξinteracttT.

In addition, we note that natural changes in the Land cover and vegetation may also be caused by natural changes in the levels of GHGs in the atmosphere. In terms of the common factors, the latter means that ξLand (natur)tT may also causally depend on ξGHG (natur)tT. Summarising what has been said, we may write a basic equation for ξLandtT as follows: 16ξLandtT=ξLand (natur)tT+ξLand (anthr)tT=SL⋅ξSoltT+OL⋅ξOrbtT+VL⋅ξVolctT+IL⋅ξinteracttT+GL⋅ξGHGtT+ξLand (anthr)tT. Notice that ξLand (natur)tT in Eq. () is influenced by ξGHGtT instead of ξGHG (natur)tT. To avoid this undesirable feature, one needs to treat ξGHG (natur)tT and ξGHG (anthr)tT as separate common factors, which could be possible if xGHG (natur)tT and xGHG (anthr)tT were available. However, climate model simulations, driven by the natural respectively anthropogenic GHG forcings, are not available.

By reasoning in a similar way, we may also formulate a corresponding basic equation for ξGHGtT, whose natural component can also be caused by the specified natural forcings and by natural changes in the land cover and vegetation: 17ξGHGtT=ξGHG (natur)tT+ξGHG (anthr)tT=SG⋅ξSoltT+OG⋅ξOrbtT+VG⋅ξVolctT+IG⋅ξinteracttT+LG⋅ξLandtT+ξGHG (anthr)tT.

Notice that Eqs. () and () together also reflect the idea that ξGHG (natur)tT and ξLand (natur)tT can be causally dependent on each other, which gives rise to a loop. Put climatologically, this loop reflects the fact that natural changes in one of these forcings can lead to subsequent changes in the same forcing by initially causing natural changes in the other forcing.

Another important comment on Eqs. () and () is that the interpretation of the interaction term ξinteracttT differs from that associated with the basic CFA model (see Eq. ). This is because within the CFA model specification, all the temperature responses are assumed to be generated by causally independent climate processes, while the SEM model specification allows for distinguishing between physically dependent and physically independent climate processes. Thus, ξinteracttT under the SEM model presented represents the overall effect of possible interactions between the solar, orbital, and volcanic forcings and the processes of anthropogenic character. Interactions between physically dependent and independent climate processes are assumed to give rise to temperature responses, whose statistical relationships within SEM models are modelled through causal inputs. Notice also that under both CFA and SEM models, it is not possible to separate the natural component of ξinteracttT from the anthropogenic because only one type of multi-forcing climate model simulations, namely xcomb, is available. Within the SEM model, it means that the causal inputs from ξinteracttT to ξLand (natural)tT and ξGHG (natural)tT are not of a purely natural origin, as we would like.

The easiest way to get an overview of the above-discussed relationships is to represent them graphically by means of a path diagram. Using the path diagram for the CFA(7, 6) model in Fig.  as a starting point, we modify it by replacing some correlations by causal inputs. The result is depicted in Fig. . A complete set of the equations, associated with the SEM model in Fig. , is given in the Supplement.

The important features of the SEM model in Fig.  are as follows:

The latent variables ξSolT, ξOrbT, ξVolcT, and ξinteractT are still exogenous and standardised variables.

Just as the previously presented CFA models, the SEM model in Fig.  is a basic model, constituting a point of departure for constructing different SEM models. This can be accomplished either by deleting some of the depicted paths or by adding new ones. For example, in order to reflect the idea that the changing climate itself can cause subsequent changes, one can introduce causal paths from xcomb and/or y back to ξLandT and/or ξGHGT or more precisely to ξLand (natur)T and ξGHG (natur)T embedded in ξLandT and ξGHGT, respectively. Note that these paths also express the idea that the internal processes can randomly contribute to natural changes in the Land and GHG forcings.

The same ideas can also be expressed by letting the observable xf variables impact ξLandT and/or ξGHGT. From the perspective of structural equation modelling, freeing paths from observed variables to latent ones entails the movement from the general standard representation of a SEM model to its alternative representation (for details of both representations, see the Supplement). The identifiability status of each initial model should be determined on a case-by-case basis.

An initial SEM model, formulated on the basis of the basic SEM model, in accordance with climatological knowledge may also be modified empirically. Useful means in providing clues to specific model expansions are modification indices (for the details see Appendix A5). The main statistical advantage of model expansions is that they improve (to various extent) the overall model fit to data. Nevertheless, such modifications should be made judiciously as they lead to a reduction in the degrees of freedom. If an initial SEM model, on the other hand, demonstrates a reasonable fit both statistically and heuristically, model simplifications might be of more interest than model expansions.

In connection with empirical data-driven modifications of SEM (and CFA) models, we would also like to emphasise that the choice of a final or tentative model should not be made exclusively on a statistical basis – any modification ought to be defensible from the climatological point of view and reflect our knowledge about both the real-world climate system and the climate model under consideration. Also, a final SEM (and CFA) model should not be taken as a correct model, even if the model was not obtained as a result of empirical modifications. When accepting a final model, we can only say that “the model may be valid” because it does not contradict our assumptions and substantive knowledge.

In Sect. 4, we suggested estimators () and () for obtaining a priori estimates of the simulated internal temperature variability. Both estimators are associated with the following assumptions: i.

The variances of δ̃frepl.i:s within an ensemble are equal.

A possible way to check the validity of the estimators is to analyse the ensemble members by means of an appropriate CFA model. To this end, two simple CFA models were formulated. Their description is given in Sect. 6.1. In Sect. 6.2, we illustrate a practical application of one of these models, thereby demonstrating practical details of fitting a CFA model.

The present paper provides a theoretical background of a new statistical framework for the evaluation of simulated responses to climate forcings against observational climate data. A key idea of the framework, comprising two groups of statistical models, is that the process of evaluation should not be limited to a single statistical model. The models suggested here are CFA and SEM models, each of which is based on the concept of latent variables. Although they are closely related to each other, there are several differences between them, which allow for a statistical modelling of climatological relationships in various ways.

The idea of using CFA and SEM models originates from D&A fingerprinting studies employing statistical models known as measurement error (ME) models (or equivalently errors-in-variables models). As a matter of fact, an ME model is a special case of a CFA model, which means that an ME model is a special case of a SEM model as well. In the present work, using this close connection between the three types of statistical models, the ME model specification has been extended first to the CFA and SEM model specifications.

The theoretical results of this work have demonstrated that both CFA and SEM models, just as ME models in D&A studies, are, first of all, capable of addressing the questions posed in D&A studies, namely the assessment of the contribution of the forcings to the temperature variability (the questions of detection and attribution) and the evaluation of climate model simulations in terms of temperature responses to forcings (the question of consistency). In addition, the extensions have provided the following advantageous possibilities:

The structure of the underlying relationships can be varied between latent temperature responses to forcings in accordance to their properties and interpretations. For example, one may assume that latent temperature responses to some forcings among those considered are mutually uncorrelated. Such restrictions are especially desirable for analysing climate data associated with the so-called weak-signal regime.